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Vectors are quantities that are defined by two things :

  • their magnitude , and
  • their direction .

In 2 dimensions, a vector is described by two components :

  • a horizontal component, \(x\) component
  • a vertical component, \(y\) component

how to write vectors on paper

these components are represented as either:

  • a column vector : \(\vec{u} = \begin{pmatrix} x \\ y \end{pmatrix}\)
  • a row vector : \(\vec{u} = \begin{pmatrix} x & y \end{pmatrix}\)

Tutorial: what vectors are, what are components and how to draw them

In this tutorial we learn what a vector is. We learn how to write vectors as both column and row vectors as well as how to represent vectors graphically; that is how to draw vectors .

Using gridded paper, draw each of the following vectors:

  • the vector \(\vec{a} = \begin{pmatrix} 5 \\ 2 \end{pmatrix}\)
  • the vector \(\vec{b} = \begin{pmatrix} -3 \\ 4 \end{pmatrix}\)
  • the vector \(\vec{c} = \begin{pmatrix} 6 \\ -3 \end{pmatrix}\)
  • the vector \(\vec{d} = \begin{pmatrix} 0 \\ 5 \end{pmatrix}\)
  • the vector \(\vec{e} = \begin{pmatrix} -2 \\ -5 \end{pmatrix}\)
  • the vector \(\vec{f} = \begin{pmatrix} -4 \\ 0 \end{pmatrix}\)
  • the vector \(\vec{g} = \begin{pmatrix} 1 \\ 6 \end{pmatrix}\)
  • the vector \(\vec{h} = \begin{pmatrix} 5 \\ 5 \end{pmatrix}\)

Note : you can download a sheet of gridded paper here.

1. For \(\vec{a} = \begin{pmatrix} 5 \\ 2 \end{pmatrix}\) we find:

how to write vectors on paper

2. For \(\vec{b} = \begin{pmatrix} -3 \\ 4 \end{pmatrix}\) we find:

how to write vectors on paper

3. For \(\vec{c} = \begin{pmatrix} 6 \\ -3 \end{pmatrix}\) we find:

how to write vectors on paper

4. For \(\vec{d} = \begin{pmatrix} 0 \\ 5 \end{pmatrix}\) we find:

how to write vectors on paper

5. For \(\vec{e} = \begin{pmatrix} -2 \\ -5 \end{pmatrix}\) we find:

how to write vectors on paper

6. For \(\vec{f} = \begin{pmatrix} -4 \\ 0 \end{pmatrix}\) we find:

how to write vectors on paper

7. For \(\vec{g} = \begin{pmatrix} 1 \\ 6 \end{pmatrix}\) we find:

how to write vectors on paper

8. For \(\vec{h} = \begin{pmatrix} 5 \\ 5 \end{pmatrix}\) we find:

how to write vectors on paper

Tutorial: How to find a vector's components

Writing your answers as column vectors, find the coordinates of each of the vectors drawn here:

1. vector \(\vec{a}\):

how to write vectors on paper

2. vector \(\vec{b}\):

how to write vectors on paper

3. vector \(\vec{c}\):

how to write vectors on paper

4. vector \(\vec{d}\):

how to write vectors on paper

5. vector \(\vec{e}\):

how to write vectors on paper

6. vector \(\vec{f}\):

how to write vectors on paper

7. vector \(\vec{g}\):

how to write vectors on paper

8. vector \(\vec{h}\):

how to write vectors on paper

Solution Without Working

  • We find vector \(\vec{a} = \begin{pmatrix} 2 \\ 5 \end{pmatrix}\).
  • We find vector \(\vec{b} = \begin{pmatrix} 5 \\ -1 \end{pmatrix}\).
  • We find vector \(\vec{c} = \begin{pmatrix} 3 \\ 2 \end{pmatrix}\).
  • We find vector \(\vec{d} = \begin{pmatrix} 0 \\ -3 \end{pmatrix}\).
  • We find vector \(\vec{e} = \begin{pmatrix} -5 \\ 0 \end{pmatrix}\).
  • We find vector \(\vec{f} = \begin{pmatrix} -4 \\ 5 \end{pmatrix}\).
  • We find vector \(\vec{g} = \begin{pmatrix} 4 \\ -6 \end{pmatrix}\).
  • We find vector \(\vec{h} = \begin{pmatrix} 4 \\ 4 \end{pmatrix}\).

Displacement Vectors

Displacement vectors are defined between 2 points and are used to describe how to get from one point to the other.

Given 2 points, \(A\begin{pmatrix}x_A,y_A\end{pmatrix}\) and \(B\begin{pmatrix}x_B,y_B\end{pmatrix}\), we can define two displacement vectors \(\vec{AB}\) and \(\vec{BA}\) where: \[\vec{AB} = \begin{pmatrix}x_B - x_A \\ y_B - y_A\end{pmatrix} \quad \text{and} \quad \vec{BA} = \begin{pmatrix}x_A - x_B \\ y_A - y_B\end{pmatrix}\]

  • \(\vec{AB}\) tell us how to get from \(A\) to \(B\)
  • \(\vec{BA}\) tell us how to get from \(B\) to \(A\)

Given the points \(A\begin{pmatrix}1,2\end{pmatrix}\), \(B\begin{pmatrix}6,5\end{pmatrix}\) and \(C\begin{pmatrix}-2,-3\end{pmatrix}\), define the vectors:

  • \(\vec{AB}\)
  • \(\vec{AC}\)
  • \(\vec{BC}\)
  • \(\vec{BA}\)

Answers with Working

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how to write vectors on paper

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  • 5.1 Vector Addition and Subtraction: Graphical Methods
  • Introduction
  • 1.1 Physics: Definitions and Applications
  • 1.2 The Scientific Methods
  • 1.3 The Language of Physics: Physical Quantities and Units
  • Section Summary
  • Key Equations
  • Concept Items
  • Critical Thinking Items
  • Performance Task
  • Multiple Choice
  • Short Answer
  • Extended Response
  • 2.1 Relative Motion, Distance, and Displacement
  • 2.2 Speed and Velocity
  • 2.3 Position vs. Time Graphs
  • 2.4 Velocity vs. Time Graphs
  • 3.1 Acceleration
  • 3.2 Representing Acceleration with Equations and Graphs
  • 4.2 Newton's First Law of Motion: Inertia
  • 4.3 Newton's Second Law of Motion
  • 4.4 Newton's Third Law of Motion
  • 5.2 Vector Addition and Subtraction: Analytical Methods
  • 5.3 Projectile Motion
  • 5.4 Inclined Planes
  • 5.5 Simple Harmonic Motion
  • 6.1 Angle of Rotation and Angular Velocity
  • 6.2 Uniform Circular Motion
  • 6.3 Rotational Motion
  • 7.1 Kepler's Laws of Planetary Motion
  • 7.2 Newton's Law of Universal Gravitation and Einstein's Theory of General Relativity
  • 8.1 Linear Momentum, Force, and Impulse
  • 8.2 Conservation of Momentum
  • 8.3 Elastic and Inelastic Collisions
  • 9.1 Work, Power, and the Work–Energy Theorem
  • 9.2 Mechanical Energy and Conservation of Energy
  • 9.3 Simple Machines
  • 10.1 Postulates of Special Relativity
  • 10.2 Consequences of Special Relativity
  • 11.1 Temperature and Thermal Energy
  • 11.2 Heat, Specific Heat, and Heat Transfer
  • 11.3 Phase Change and Latent Heat
  • 12.1 Zeroth Law of Thermodynamics: Thermal Equilibrium
  • 12.2 First law of Thermodynamics: Thermal Energy and Work
  • 12.3 Second Law of Thermodynamics: Entropy
  • 12.4 Applications of Thermodynamics: Heat Engines, Heat Pumps, and Refrigerators
  • 13.1 Types of Waves
  • 13.2 Wave Properties: Speed, Amplitude, Frequency, and Period
  • 13.3 Wave Interaction: Superposition and Interference
  • 14.1 Speed of Sound, Frequency, and Wavelength
  • 14.2 Sound Intensity and Sound Level
  • 14.3 Doppler Effect and Sonic Booms
  • 14.4 Sound Interference and Resonance
  • 15.1 The Electromagnetic Spectrum
  • 15.2 The Behavior of Electromagnetic Radiation
  • 16.1 Reflection
  • 16.2 Refraction
  • 16.3 Lenses
  • 17.1 Understanding Diffraction and Interference
  • 17.2 Applications of Diffraction, Interference, and Coherence
  • 18.1 Electrical Charges, Conservation of Charge, and Transfer of Charge
  • 18.2 Coulomb's law
  • 18.3 Electric Field
  • 18.4 Electric Potential
  • 18.5 Capacitors and Dielectrics
  • 19.1 Ohm's law
  • 19.2 Series Circuits
  • 19.3 Parallel Circuits
  • 19.4 Electric Power
  • 20.1 Magnetic Fields, Field Lines, and Force
  • 20.2 Motors, Generators, and Transformers
  • 20.3 Electromagnetic Induction
  • 21.1 Planck and Quantum Nature of Light
  • 21.2 Einstein and the Photoelectric Effect
  • 21.3 The Dual Nature of Light
  • 22.1 The Structure of the Atom
  • 22.2 Nuclear Forces and Radioactivity
  • 22.3 Half Life and Radiometric Dating
  • 22.4 Nuclear Fission and Fusion
  • 22.5 Medical Applications of Radioactivity: Diagnostic Imaging and Radiation
  • 23.1 The Four Fundamental Forces
  • 23.2 Quarks
  • 23.3 The Unification of Forces
  • A | Reference Tables

Section Learning Objectives

By the end of this section, you will be able to do the following:

  • Describe the graphical method of vector addition and subtraction
  • Use the graphical method of vector addition and subtraction to solve physics problems

Teacher Support

The learning objectives in this section will help your students master the following standards:

  • (E) develop and interpret free-body force diagrams.

Section Key Terms

The graphical method of vector addition and subtraction.

Recall that a vector is a quantity that has magnitude and direction. For example, displacement, velocity, acceleration, and force are all vectors. In one-dimensional or straight-line motion, the direction of a vector can be given simply by a plus or minus sign. Motion that is forward, to the right, or upward is usually considered to be positive (+); and motion that is backward, to the left, or downward is usually considered to be negative (−).

In two dimensions, a vector describes motion in two perpendicular directions, such as vertical and horizontal. For vertical and horizontal motion, each vector is made up of vertical and horizontal components. In a one-dimensional problem, one of the components simply has a value of zero. For two-dimensional vectors, we work with vectors by using a frame of reference such as a coordinate system. Just as with one-dimensional vectors, we graphically represent vectors with an arrow having a length proportional to the vector’s magnitude and pointing in the direction that the vector points.

[BL] [OL] Review vectors and free body diagrams. Recall how velocity, displacement and acceleration vectors are represented.

Figure 5.2 shows a graphical representation of a vector; the total displacement for a person walking in a city. The person first walks nine blocks east and then five blocks north. Her total displacement does not match her path to her final destination. The displacement simply connects her starting point with her ending point using a straight line, which is the shortest distance. We use the notation that a boldface symbol, such as D , stands for a vector. Its magnitude is represented by the symbol in italics, D , and its direction is given by an angle represented by the symbol θ . θ . Note that her displacement would be the same if she had begun by first walking five blocks north and then walking nine blocks east.

Tips For Success

In this text, we represent a vector with a boldface variable. For example, we represent a force with the vector F , which has both magnitude and direction. The magnitude of the vector is represented by the variable in italics, F , and the direction of the variable is given by the angle θ . θ .

The head-to-tail method is a graphical way to add vectors. The tail of the vector is the starting point of the vector, and the head (or tip) of a vector is the pointed end of the arrow. The following steps describe how to use the head-to-tail method for graphical vector addition .

  • If there are more than two vectors, continue to add the vectors head-to-tail as described in step 2. In this example, we have only two vectors, so we have finished placing arrows tip to tail.
  • To find the magnitude of the resultant, measure its length with a ruler. When we deal with vectors analytically in the next section, the magnitude will be calculated by using the Pythagorean theorem.
  • To find the direction of the resultant, use a protractor to measure the angle it makes with the reference direction (in this case, the x -axis). When we deal with vectors analytically in the next section, the direction will be calculated by using trigonometry to find the angle.

[AL] Ask two students to demonstrate pushing a table from two different directions. Ask students what they feel the direction of resultant motion will be. How would they represent this graphically? Recall that a vector’s magnitude is represented by the length of the arrow. Demonstrate the head-to-tail method of adding vectors, using the example given in the chapter. Ask students to practice this method of addition using a scale and a protractor.

[BL] [OL] [AL] Ask students if anything changes by moving the vector from one place to another on a graph. How about the order of addition? Would that make a difference? Introduce negative of a vector and vector subtraction.

Watch Physics

Visualizing vector addition examples.

This video shows four graphical representations of vector addition and matches them to the correct vector addition formula.

  • Yes, if we add the same two vectors in a different order it will still give the same resultant vector.
  • No, the resultant vector will change if we add the same vectors in a different order.

Vector subtraction is done in the same way as vector addition with one small change. We add the first vector to the negative of the vector that needs to be subtracted. A negative vector has the same magnitude as the original vector, but points in the opposite direction (as shown in Figure 5.6 ). Subtracting the vector B from the vector A , which is written as A − B , is the same as A + (− B ). Since it does not matter in what order vectors are added, A − B is also equal to (− B ) + A . This is true for scalars as well as vectors. For example, 5 – 2 = 5 + (−2) = (−2) + 5.

Global angles are calculated in the counterclockwise direction. The clockwise direction is considered negative. For example, an angle of 30 ∘ 30 ∘ south of west is the same as the global angle 210 ∘ , 210 ∘ , which can also be expressed as −150 ∘ −150 ∘ from the positive x -axis.

Using the Graphical Method of Vector Addition and Subtraction to Solve Physics Problems

Now that we have the skills to work with vectors in two dimensions, we can apply vector addition to graphically determine the resultant vector , which represents the total force. Consider an example of force involving two ice skaters pushing a third as seen in Figure 5.7 .

In problems where variables such as force are already known, the forces can be represented by making the length of the vectors proportional to the magnitudes of the forces. For this, you need to create a scale. For example, each centimeter of vector length could represent 50 N worth of force. Once you have the initial vectors drawn to scale, you can then use the head-to-tail method to draw the resultant vector. The length of the resultant can then be measured and converted back to the original units using the scale you created.

You can tell by looking at the vectors in the free-body diagram in Figure 5.7 that the two skaters are pushing on the third skater with equal-magnitude forces, since the length of their force vectors are the same. Note, however, that the forces are not equal because they act in different directions. If, for example, each force had a magnitude of 400 N, then we would find the magnitude of the total external force acting on the third skater by finding the magnitude of the resultant vector. Since the forces act at a right angle to one another, we can use the Pythagorean theorem. For a triangle with sides a, b, and c, the Pythagorean theorem tells us that

Applying this theorem to the triangle made by F 1 , F 2 , and F tot in Figure 5.7 , we get

Note that, if the vectors were not at a right angle to each other ( 90 ∘ ( 90 ∘ to one another), we would not be able to use the Pythagorean theorem to find the magnitude of the resultant vector. Another scenario where adding two-dimensional vectors is necessary is for velocity, where the direction may not be purely east-west or north-south, but some combination of these two directions. In the next section, we cover how to solve this type of problem analytically. For now let’s consider the problem graphically.

Worked Example

Adding vectors graphically by using the head-to-tail method: a woman takes a walk.

Use the graphical technique for adding vectors to find the total displacement of a person who walks the following three paths (displacements) on a flat field. First, she walks 25 m in a direction 49 ∘ 49 ∘ north of east. Then, she walks 23 m heading 15 ∘ 15 ∘ north of east. Finally, she turns and walks 32 m in a direction 68 ∘ 68 ∘ south of east.

Graphically represent each displacement vector with an arrow, labeling the first A , the second B , and the third C . Make the lengths proportional to the distance of the given displacement and orient the arrows as specified relative to an east-west line. Use the head-to-tail method outlined above to determine the magnitude and direction of the resultant displacement, which we’ll call R .

(1) Draw the three displacement vectors, creating a convenient scale (such as 1 cm of vector length on paper equals 1 m in the problem), as shown in Figure 5.8 .

(2) Place the vectors head to tail, making sure not to change their magnitude or direction, as shown in Figure 5.9 .

(3) Draw the resultant vector R from the tail of the first vector to the head of the last vector, as shown in Figure 5.10 .

(4) Use a ruler to measure the magnitude of R , remembering to convert back to the units of meters using the scale. Use a protractor to measure the direction of R . While the direction of the vector can be specified in many ways, the easiest way is to measure the angle between the vector and the nearest horizontal or vertical axis. Since R is south of the eastward pointing axis (the x -axis), we flip the protractor upside down and measure the angle between the eastward axis and the vector, as illustrated in Figure 5.11 .

In this case, the total displacement R has a magnitude of 50 m and points 7 ∘ 7 ∘ south of east. Using its magnitude and direction, this vector can be expressed as

The head-to-tail graphical method of vector addition works for any number of vectors. It is also important to note that it does not matter in what order the vectors are added. Changing the order does not change the resultant. For example, we could add the vectors as shown in Figure 5.12 , and we would still get the same solution.

[BL] [OL] [AL] Ask three students to enact the situation shown in Figure 5.8 . Recall how these forces can be represented in a free-body diagram. Giving values to these vectors, show how these can be added graphically.

Subtracting Vectors Graphically: A Woman Sailing a Boat

A woman sailing a boat at night is following directions to a dock. The instructions read to first sail 27.5 m in a direction 66.0 ∘ 66.0 ∘ north of east from her current location, and then travel 30.0 m in a direction 112 ∘ 112 ∘ north of east (or 22.0 ∘ 22.0 ∘ west of north). If the woman makes a mistake and travels in the opposite direction for the second leg of the trip, where will she end up? The two legs of the woman’s trip are illustrated in Figure 5.13 .

We can represent the first leg of the trip with a vector A , and the second leg of the trip that she was supposed to take with a vector B . Since the woman mistakenly travels in the opposite direction for the second leg of the journey, the vector for second leg of the trip she actually takes is − B . Therefore, she will end up at a location A + (− B ), or A − B . Note that − B has the same magnitude as B (30.0 m), but is in the opposite direction, 68 ∘ ( 180 ∘ − 112 ∘ ) 68 ∘ ( 180 ∘ − 112 ∘ ) south of east, as illustrated in Figure 5.14 .

We use graphical vector addition to find where the woman arrives A + (− B ).

(1) To determine the location at which the woman arrives by accident, draw vectors A and − B .

(2) Place the vectors head to tail.

(3) Draw the resultant vector R .

(4) Use a ruler and protractor to measure the magnitude and direction of R .

These steps are demonstrated in Figure 5.15 .

In this case

Because subtraction of a vector is the same as addition of the same vector with the opposite direction, the graphical method for subtracting vectors works the same as for adding vectors.

Adding Velocities: A Boat on a River

A boat attempts to travel straight across a river at a speed of 3.8 m/s. The river current flows at a speed v river of 6.1 m/s to the right. What is the total velocity and direction of the boat? You can represent each meter per second of velocity as one centimeter of vector length in your drawing.

We start by choosing a coordinate system with its x-axis parallel to the velocity of the river. Because the boat is directed straight toward the other shore, its velocity is perpendicular to the velocity of the river. We draw the two vectors, v boat and v river , as shown in Figure 5.16 .

Using the head-to-tail method, we draw the resulting total velocity vector from the tail of v boat to the head of v river .

By using a ruler, we find that the length of the resultant vector is 7.2 cm, which means that the magnitude of the total velocity is

By using a protractor to measure the angle, we find θ = 32.0 ∘ . θ = 32.0 ∘ .

If the velocity of the boat and river were equal, then the direction of the total velocity would have been 45°. However, since the velocity of the river is greater than that of the boat, the direction is less than 45° with respect to the shore, or x axis.

Teacher Demonstration

Plot the way from the classroom to the cafeteria (or any two places in the school on the same level). Ask students to come up with approximate distances. Ask them to do a vector analysis of the path. What is the total distance travelled? What is the displacement?

Practice Problems

Virtual physics, vector addition.

In this simulation , you will experiment with adding vectors graphically. Click and drag the red vectors from the Grab One basket onto the graph in the middle of the screen. These red vectors can be rotated, stretched, or repositioned by clicking and dragging with your mouse. Check the Show Sum box to display the resultant vector (in green), which is the sum of all of the red vectors placed on the graph. To remove a red vector, drag it to the trash or click the Clear All button if you wish to start over. Notice that, if you click on any of the vectors, the | R | | R | is its magnitude, θ θ is its direction with respect to the positive x -axis, R x is its horizontal component, and R y is its vertical component. You can check the resultant by lining up the vectors so that the head of the first vector touches the tail of the second. Continue until all of the vectors are aligned together head-to-tail. You will see that the resultant magnitude and angle is the same as the arrow drawn from the tail of the first vector to the head of the last vector. Rearrange the vectors in any order head-to-tail and compare. The resultant will always be the same.

Grasp Check

True or False—The more long, red vectors you put on the graph, rotated in any direction, the greater the magnitude of the resultant green vector.

Check Your Understanding

  • backward and to the left
  • backward and to the right
  • forward and to the right
  • forward and to the left

True or False—A person walks 2 blocks east and 5 blocks north. Another person walks 5 blocks north and then two blocks east. The displacement of the first person will be more than the displacement of the second person.

Use the Check Your Understanding questions to assess whether students achieve the learning objectives for this section. If students are struggling with a specific objective, the Check Your Understanding will help identify which objective is causing the problem and direct students to the relevant content.

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  • Authors: Paul Peter Urone, Roger Hinrichs
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  • Publication date: Mar 26, 2020
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  • Book URL: https://openstax.org/books/physics/pages/1-introduction
  • Section URL: https://openstax.org/books/physics/pages/5-1-vector-addition-and-subtraction-graphical-methods

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Mathematics LibreTexts

2.2: Vector Equations and Spans

  • Last updated
  • Save as PDF
  • Page ID 70187

  • Dan Margalit & Joseph Rabinoff
  • Georgia Institute of Technology
  • Understand the equivalence between a system of linear equations and a vector equation.
  • Learn the definition of \(\text{Span}\{x_1,x_2,\ldots,x_k\}\text{,}\) and how to draw pictures of spans.
  • Recipe: solve a vector equation using augmented matrices / decide if a vector is in a span.
  • Pictures: an inconsistent system of equations, a consistent system of equations, spans in \(\mathbb{R}^2\) and \(\mathbb{R}^3\).
  • Vocabulary word: vector equation .
  • Essential vocabulary word: span .

Vector Equations

An equation involving vectors with \(n\) coordinates is the same as \(n\) equations involving only numbers. For example, the equation

\[x\left(\begin{array}{c}1\\2\\6\end{array}\right) +y\left(\begin{array}{c}-1\\-2\\-1\end{array}\right)=\left(\begin{array}{c}8\\16\\3\end{array}\right)\nonumber\]

simplifies to

\[\left(\begin{array}{c}x\\2x\\6x\end{array}\right)+\left(\begin{array}{c}-y\\-2y\\-y\end{array}\right)=\left(\begin{array}{c}8\\16\\3\end{array}\right)\quad\text{or}\quad\left(\begin{array}{c}x-y\\2x-2y\\6x-y\end{array}\right)=\left(\begin{array}{c}8\\16\\3\end{array}\right).\nonumber\]

For two vectors to be equal, all of their coordinates must be equal, so this is just the system of linear equations

\[\left\{\begin{array}{rrrrc}x &-& y &=& 8\\ 2x &-& 2y &=& 16\\ 6x &-& y &=& 3.\end{array}\right.\nonumber\]

Definition \(\PageIndex{1}\): Vector Equation

A vector equation is an equation involving a linear combination of vectors with possibly unknown coefficients.

Note \(\PageIndex{1}\)

Asking whether or not a vector equation has a solution is the same as asking if a given vector is a linear combination of some other given vectors.

For example the vector equation above is asking if the vector \((8,16,3)\) is a linear combination of the vectors \((1,2,6)\) and \((-1,2,-1)\).

The thing we really care about is solving systems of linear equations, not solving vector equations. The whole point of vector equations is that they give us a different, and more geometric, way of viewing systems of linear equations.

Note \(\PageIndex{2}\): A Picture of a Consistent System

Below we will show that the above system of equations is consistent. Equivalently, this means that the above vector equation has a solution. In other words, there is a linear combination of \((1,2,6)\) and \((-1,2,-1)\) that equals \((8,16,3)\). We can visualize the last statement geometrically. Therefore, the following Figure \(\PageIndex{1}\) gives a picture of a consistent system of equations . Compare with Figure \(\PageIndex{2}\), which shows a picture of an inconsistent system.

clipboard_ef1824517b4680c6e21b27818e45b8b43.png

In order to actually solve the vector equation

\[x\color{Red}{\left(\begin{array}{c}1\\2\\6\end{array}\right)} \color{black}{+y}\color{Green}{\left(\begin{array}{c}-1\\-2\\-1\end{array}\right)}\color{black}{=}\color{blue}{\left(\begin{array}{c}8\\16\\3\end{array}\right)}\color{black}{,}\nonumber\]

one has to solve the system of linear equations

\[\left\{\begin{array}{rrrrc} x &-& y &=& 8\\ 2x &-& 2y &=& 16\\ 6x &-& y &=& 3.\end{array}\right. \nonumber\]

This means forming the augmented matrix

\[\left(\begin{array}{cc|c}\color{Red}{1}&\color{Green}{-1}&\color{blue}{8} \\ \color{Red}{2}&\color{Green}{-2}&\color{blue}{16}\\ \color{Red}{6}&\color{Green}{-1} &\color{blue}{3}\end{array}\right)\nonumber\]

and row reducing. Note that the columns of the augmented matrix are the vectors from the original vector equation , so it is not actually necessary to write the system of equations: one can go directly from the vector equation to the augmented matrix by “smooshing the vectors together”. In Example \(\PageIndex{1}\) we carry out the row reduction and find the solution.

Example \(\PageIndex{1}\)

Is \(\left(\begin{array}{c}8\\16\\3\end{array}\right)\) a linear combination of \(\left(\begin{array}{c}1\\2\\6\end{array}\right)\) and \(\left(\begin{array}{c}-1\\-2\\-1\end{array}\right)\)?

As discussed above, this question boils down to a row reduction:

\[\left(\begin{array}{cc|c} 1&-1&8 \\ 2&-2&16 \\ 6&-1&3\end{array}\right) \quad\xrightarrow{\text{RREF}}\quad \left(\begin{array}{cc|c} 1&0&-1 \\ 0&1&-9 \\ 0&0&0\end{array}\right).\nonumber\]

From this we see that the equation is consistent, and the solution is \(x=-1\) and \(y=-9\). We conclude that \(\left(\begin{array}{c}8\\16\\3\end{array}\right)\) is indeed a linear combination of \(\left(\begin{array}{c}1\\2\\6\end{array}\right)\) and \(\left(\begin{array}{c}-1\\-2\\-1\end{array}\right)\), with coefficients \(-1\) and \(-9\text{:}\)

\[-\left(\begin{array}{c}1\\2\\6\end{array}\right)-9\left(\begin{array}{c}-1\\-2\\-1\end{array}\right)=\left(\begin{array}{c}8\\6\\3\end{array}\right).\nonumber\]

Recipe: Solving a Vector Equation

In general, the vector equation

\[ x_1v_1 + x_2v_2 + \cdots + x_kv_k = b \nonumber \]

where \(v_1,v_2,\ldots,v_k,\,b\) are vectors in \(\mathbb{R}^n\) and \(x_1,x_2,\ldots,x_k\) are unknown scalars, has the same solution set as the linear system with augmented matrix

\[\left(\begin{array}{cccc|c} | & |&\quad &|&| \\ v_1 &v_2 &\cdots &v_k &b \\ |&|&\quad &|&|\end{array}\right)\nonumber\]

whose columns are the \(v_i\)’s and the \(b\)’s.

Now we have three equivalent ways of thinking about a linear system:

  • As a system of equations: \[\left\{\begin{array}{rrrrrrr} 2x_1 &+& 3x_2 &-& 2x_3 &=& 7\\ x_1 &-& x_2 &-& 3x_3 &=& 5\end{array}\right.\nonumber\]
  • As an augmented matrix: \[\left(\begin{array}{ccc|c} 2&3&-2&7 \\ 1&-1&-3&5\end{array}\right)\nonumber\]
  • As a vector equation (\(x_1v_1 + x_2v_2 + \cdots + x_nv_n = b\)): \[x_{1}\left(\begin{array}{c}2\\1\end{array}\right)+x_{2}\left(\begin{array}{c}3\\-1\end{array}\right)+x_{3}\left(\begin{array}{c}-2\\-3\end{array}\right)=\left(\begin{array}{c}7\\5\end{array}\right)\nonumber\]

The third is geometric in nature: it lends itself to drawing pictures.

It will be important to know what are all linear combinations of a set of vectors \(v_1,v_2,\ldots,v_k\) in \(\mathbb{R}^n\). In other words, we would like to understand the set of all vectors \(b\) in \(\mathbb{R}^n\) such that the vector equation (in the unknowns \(x_1,x_2,\ldots,x_k\))

has a solution (i.e. is consistent).

Definition \(\PageIndex{2}\): Span

Let \(v_1,v_2,\ldots,v_k\) be vectors in \(\mathbb{R}^n\). The span of \(v_1,v_2,\ldots,v_k\) is the collection of all linear combinations of \(v_1,v_2,\ldots,v_k\text{,}\) and is denoted \(\text{Span}\{v_1,v_2,\ldots,v_k\}\). In symbols:

\[ \text{Span}\{v_1,v_2,\ldots,v_k\} = \bigl\{x_1v_1 + x_2v_2 + \cdots + x_kv_k \mid x_1,x_2,\ldots,x_k \text{ in }\mathbb{R}\bigr\} \nonumber \]

We also say that \(\text{Span}\{v_1,v_2,\ldots,v_k\}\) is the subset spanned by or generated by the vectors \(v_1,v_2,\ldots,v_k\).

The above definition, Definition \(\PageIndex{2}\) is the first of several essential definitions that we will see in this textbook. They are essential in that they form the essence of the subject of linear algebra: learning linear algebra means (in part) learning these definitions. All of the definitions are important, but it is essential that you learn and understand the definitions marked as such.

Note \(\PageIndex{3}\): Set Builder Notation

The notation

\[ \bigl\{x_1v_1 + x_2v_2 + \cdots + x_kv_k \mid x_1,x_2,\ldots,x_k \text{ in }\mathbb{R}\bigr\} \nonumber \]

reads as: “the set of all things of the form \(x_1v_1 + x_2v_2 + \cdots + x_kv_k\) such that \(x_1,x_2,\ldots,x_k\) are in \(\mathbb{R}\).” The vertical line is “such that”; everything to the left of it is “the set of all things of this form”, and everything to the right is the condition that those things must satisfy to be in the set. Specifying a set in this way is called set builder notation .

All mathematical notation is only shorthand: any sequence of symbols must translate into a usual sentence.

Note \(\PageIndex{4}\): Three characterizations of consistency

Now we have three equivalent ways of making the same statement:

  • A vector \(b\) is in the span of \(v_1,v_2,\ldots,v_k\).
  • The vector equation \[x_1 v_1 +x_2 v_2 +\cdots +x_k v_k =b\nonumber\] has a solution.
  • The linear system with augmented matrix \[\left(\begin{array}{cccc|c} |&|&\quad &|&| \\ v_1 &v_2 &\cdots &v_k &b \\ |&|&\quad &|&| \end{array}\right)\nonumber\]  is consistent.

Equivalent means that, for any given list of vectors \(v_1,v_2,\ldots,v_k,\,b\text{,}\) either all three statements are true, or all three statements are false.

clipboard_e469573233e49c52bca5c4fe06e4beb9c.png

Pictures of Spans.

Drawing a picture of \(\text{Span}\{v_1,v_2,\ldots,v_k\}\) is the same as drawing a picture of all linear combinations of \(v_1,v_2,\ldots,v_k\).

clipboard_ea1aaaaf4e548a0ef6a4cd44b2c609a9c.png

Figure  \(\PageIndex{3}\): Pictures of spans in \(\mathbb{R}^2\).

clipboard_e953598b938f4e6f1ac1ff2669909ef88.png

Example \(\PageIndex{2}\): Interactive: Span of two vectors in \(\mathbb{R}^2\)

clipboard_e0e44c66998c05c23b00e32b920eb4748.png

Example \(\PageIndex{3}\): Interactive: Span of two vectors in \(\mathbb{R}^3\)

clipboard_ebd384f245878d05a07041a9cc87ba196.png

Example \(\PageIndex{4}\): Interactive: Span of three vectors in \(\mathbb{R}^3\)

clipboard_ed0cf11d202082456e9259f09cac52eac.png

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Rhett Allain

How do you represent vectors?

Recently , I was talking about vectors. At that time, I had to stop and recall how I had been representing vectors. Ideally, I should stick with the same notation I used in Basics: Vectors and Vector Addition . But let me go over the different ways you could represent a vector.

Maybe this is too obvious, but it had to be said. You can represent vectors by drawing them. In fact, this is very useful conceptually - but maybe not too useful for calculations. When a vector is represented graphically, its magnitude is represented by the length of an arrow and its direction is represented by the direction of the arrow. Here is an example:

how to write vectors on paper

I think the biggest negative to this representation (other than being difficult to get numerical answers for adding) is that it is not too easy to represent in 3-dimensions. For the following representations, I will try to relate them to the graphical representation.

In algebra-based courses, maybe this format is popular. Basically, you just give the magnitude of the vector and the angle (from the positive x-axis) that the vector is pointing. Here is an example (using the same vector from before):

how to write vectors on paper

And in magnitude-direction format, it would be:

how to write vectors on paper

I am not too found of this format. First, if you want to add vectors, you need to find components. Second, students often get confused with this angle always being measured from the same axis (it doesn't have to be the x-axis, that is just what is common). Oh, if you want to do this for a 3-D vector, it really isn't worth it. You would need two angles. Well, in some cases it might be worth it.

With the component method, the idea is to just give the amount the vector is in each of the coordinate directions. Here is an example.

how to write vectors on paper

Hold on. I am not finished. Yes, I wrote these components as vectors so that:

how to write vectors on paper

Often you will see textbooks sort of stop here. In this case they may say something like:

how to write vectors on paper

It is important to realize that this notation is NOT the magnitude of the vector F x and F y . The magnitude of a vector has to be a positive number. To really use these, you need unit vectors. This is what they look like:

how to write vectors on paper

The little ^ over the x means that it is a unit vector. A unit vector is a vector that has a magnitude of 1 with no units. This means that the F x vector could be written as:

how to write vectors on paper

And maybe now you can see why that negative sign is important. The vector F x is in the opposite direction as the x-hat vector and that is why you need a negative sign. So, using this notation, you could write the vector F as:

how to write vectors on paper

Some textbooks like the you i, and j instead of x and y - this would look like:

how to write vectors on paper

Same thing, different looks. Don't forget units though. Vectors have units, if you leave them off you are probably a mathematician (just kidding). Also, this notation can be expanded to three dimensions by adding a z-hat or k-hat component. Another nice thing is that these vectors are all set up and ready to add. If you have a vector in component notation you are ready to rock.

I guess the reason textbooks use the magnitude-direction format some is that it may be easier to relate to real life. In real life, I would measure the magnitude and direction of a force and then have to calculate the components.

I really like the physics textbook Matter and Interactions by Ruth Chabay and Bruce Sherwood. The way that textbook consistently represents vectors is as:

how to write vectors on paper

I like this notation. It is short and it emphasizes the components as well as the idea that all forces are 3-dimensional. The short thing is really good for lazy people like me. Also, it matches up really nicely with vectors in vpython . Here is how I would write that vector in vpython:

how to write vectors on paper

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This is a vector:

A vector has magnitude (size) and direction :

The length of the line shows its magnitude and the arrowhead points in the direction.

Play with one here:

We can add two vectors by joining them head-to-tail:

And it doesn't matter which order we add them, we get the same result:

Example: A plane is flying along, pointing North, but there is a wind coming from the North-West.

The two vectors (the velocity caused by the propeller, and the velocity of the wind) result in a slightly slower ground speed heading a little East of North.

If you watched the plane from the ground it would seem to be slipping sideways a little.

Have you ever seen that happen? Maybe you have seen birds struggling against a strong wind that seem to fly sideways. Vectors help explain that.

Velocity , acceleration , force and many other things are vectors.

Subtracting

We can also subtract one vector from another:

  • first we reverse the direction of the vector we want to subtract,
  • then add them as usual:

A vector is often written in bold , like a or b .

Calculations

Now ... how do we do the calculations?

The most common way is to first break up vectors into x and y parts, like this:

The vector a is broken up into the two vectors a x and a y

(We see later how to do this.)

Adding Vectors

We can then add vectors by adding the x parts and adding the y parts :

The vector (8, 13) and the vector (26, 7) add up to the vector (34, 20)

Example: add the vectors a = (8, 13) and b = (26, 7)

c = (8, 13) + (26, 7) = (8+26, 13+7) = (34, 20)

When we break up a vector like that, each part is called a component :

Subtracting Vectors

To subtract, first reverse the vector we want to subtract, then add.

Example: subtract k = (4, 5) from v = (12, 2)

a = v + − k

a = (12, 2) + −(4, 5) = (12, 2) + (−4, −5) = (12−4, 2−5) = (8, −3)

Magnitude of a Vector

The magnitude of a vector is shown by two vertical bars on either side of the vector:

OR it can be written with double vertical bars (so as not to confuse it with absolute value):

We use Pythagoras' theorem to calculate it:

| a | = √( x 2 + y 2 )

Example: what is the magnitude of the vector b = (6, 8) ?

| b | = √( 6 2 + 8 2 ) = √(36+64) = √100 = 10

A vector with magnitude 1 is called a Unit Vector .

Vector vs Scalar

A scalar has magnitude (size) only .

Scalar: just a number (like 7 or −0.32) ... definitely not a vector.

A vector has magnitude and direction , and is often written in bold , so we know it is not a scalar:

  • so c is a vector, it has magnitude and direction
  • but c is just a value, like 3 or 12.4

Example: k b is actually the scalar k times the vector b .

Multiplying a vector by a scalar.

When we multiply a vector by a scalar it is called "scaling" a vector, because we change how big or small the vector is.

Example: multiply the vector m = (7, 3) by the scalar 3

It still points in the same direction, but is 3 times longer

(And now you know why numbers are called "scalars", because they "scale" the vector up or down.)

Multiplying a Vector by a Vector (Dot Product and Cross Product)

How do we multiply two vectors together? There is more than one way!

  • The scalar or Dot Product (the result is a scalar).
  • The vector or Cross Product (the result is a vector).

(Read those pages for more details.)

More Than 2 Dimensions

Vectors also work perfectly well in 3 or more dimensions:

Example: add the vectors a = (3, 7, 4) and b = (2, 9, 11)

c = (3, 7, 4) + (2, 9, 11) = (3+2, 7+9, 4+11) = (5, 16, 15)

Example: what is the magnitude of the vector w = (1, −2, 3) ?

| w | = √( 1 2 + (−2) 2 + 3 2 ) = √(1+4+9) = √14

Here is an example with 4 dimensions (but it is hard to draw!):

Example: subtract (1, 2, 3, 4) from (3, 3, 3, 3)

(3, 3, 3, 3) + −(1, 2, 3, 4) = (3, 3, 3, 3) + (−1,−2,−3,−4) = (3−1, 3−2, 3−3, 3−4) = (2, 1, 0, −1)

Magnitude and Direction

We may know a vector's magnitude and direction, but want its x and y lengths (or vice versa):

You can read how to convert them at Polar and Cartesian Coordinates , but here is a quick summary:

Sam and Alex are pulling a box.

  • Sam pulls with 200 Newtons of force at 60°
  • Alex pulls with 120 Newtons of force at 45° as shown

What is the combined force , and its direction?

Let us add the two vectors head to tail:

First convert from polar to Cartesian (to 2 decimals):

Sam's Vector:

  • x = r × cos( θ ) = 200 × cos(60°) = 200 × 0.5 = 100
  • y = r × sin( θ ) = 200 × sin(60°) = 200 × 0.8660 = 173.21

Alex's Vector:

  • x = r × cos( θ ) = 120 × cos(−45°) = 120 × 0.7071 = 84.85
  • y = r × sin( θ ) = 120 × sin(−45°) = 120 × -0.7071 = −84.85

Now we have:

(100,173.21) + (84.85, −84.85) = (184.85, 88.36)

That answer is valid, but let's convert back to polar as the question was in polar:

  • r = √ ( x 2 + y 2 ) = √ ( 184.85 2 + 88.36 2 ) = 204.88
  • θ = tan -1 ( y / x ) = tan -1 ( 88.36 / 184.85 ) = 25.5°

They might get a better result if they were shoulder-to-shoulder!

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Column Vector

Here we will learn about column vectors, including how to write a column vector and how to draw a diagram of a column vector.

There are also vector worksheets based on Edexcel, AQA and OCR exam questions, along with further guidance on where to go next if you’re still stuck.

What is a column vector?

A column vector is a way of writing a vector which gives information about the vector. It is split into a horizontal component and a vertical component.

There is a horizontal component, also known as the \textbf{x} component This is the top number in the column vector and tells us how many spaces to the right or left to move.

If the number is positive , the direction is to the right .  

If the number is negative , the direction is to the left .

There is a vertical component , also known as the \textbf{y} component. This is the bottom number in the column vector and tells us how many spaces up or down to move.  

If the number is positive , the direction is upwards .  

If the number is negative , the direction is downwards .

Vector \textbf{a} can be written as the column vector \begin{pmatrix} \; 3 \;\\ \; 2 \; \end{pmatrix}

Notice the horizontal component and the vertical component make a right-angled triangle.

Vector \textbf{b} can be written as the column vector \begin{pmatrix} \; 3 \;\\ \; -4 \; \end{pmatrix}

What is a column vector?

How to write a column vector

In order to write a vector as a column vector:

Work out the horizontal component ( \textbf{x} component).

Work out the vertical component ( \textbf{y} component).

Write the column vector.

How to write a column vector

Column vector worksheet

Get your free column vector worksheet of 20+ questions and answers. Includes reasoning and applied questions.

Related lessons on vectors

Column vector is part of our series of lessons to support revision on vectors . You may find it helpful to start with the main vectors lesson for a summary of what to expect, or use the step by step guides below for further detail on individual topics. Other lessons in this series include:

  • Magnitude of a vector  
  • Vector notation
  • Vector multiplication
  • Vector addition
  • Vector subtraction
  • Vector problems

Column vector examples

Example 1: write a column vector.

Write vector \textbf{a} as a column vector.

how to write vectors on paper

From the starting point of the vector, draw a horizontal line.

This line is 4 squares to the right.

how to write vectors on paper

2 Work out the vertical component ( \textbf{y} component).

From the end of the horizontal component, draw a vertical line to the end of the vector.

This line is 3 squares up.

how to write vectors on paper

3 Write the column vector.

Write the horizontal component and the vertical component in a column vector. 

Vector \textbf{a} as a column vector is:

Example 2: write a column vector

Write vector \textbf{b} as a column vector.

From the starting point of the vector, draw a horizontal line. 

This line is 1 square down.

Vector \textbf{b} as a column vector is:

\textbf{b}= \begin{pmatrix} \; 4 \;\\ \; -1 \; \end{pmatrix}

Example 3: write a column vector

Write vector \textbf{v} as a column vector.

From the starting point of the vector, draw a horizontal line.  We are trying to make a right-angled triangle. 

This line is 2 squares to the left.

Vector \textbf{v} as a column vector is:

\textbf{v}= \begin{pmatrix} \; -2 \;\\ \; -1 \; \end{pmatrix}

How to draw a vector using a column vector

In order to draw a diagram of a column vector:

Draw the horizontal component ( \textbf{x} component).

Draw the vertical component ( \textbf{y} component).

Draw the vector.

How to draw a vector using a column vector

Drawing a diagram of a column vector examples

Example 4: draw a diagram of the column vector.

Draw a diagram of the column vector \begin{pmatrix} \; 2 \;\\ \; 5 \; \end{pmatrix}

On the grid choose a starting point and draw the horizontal component

The top number is 2 so we draw a line 2 squares to the right .

From the end of the horizontal component, draw the vertical component.

The bottom number is 5 so we draw a line 5 squares up.

Join up the starting point and the end point and remember to put the direction arrow on the line.

Example 5: draw a diagram of the column vector

Draw a diagram of the column vector \begin{pmatrix} \; -3 \;\\ \; 1 \; \end{pmatrix}

The top number is -3 so we draw a line 3 squares to the left .

From the end of the horizontal component, draw the vertical component. 

The bottom number is 1 so we draw a line 1 square up.

Example 6: draw a diagram of the column vector

Draw a diagram of the column vector \begin{pmatrix} \; -6 \;\\ \; -1 \; \end{pmatrix}

The top number is -6 so we draw a line 6 squares to the left.

The bottom number is -1 so we draw a line 1 square down.

Common misconceptions

  • Make sure the signs are correct

Remember: If the top number is positive, the direction is to the right.  If the top number is negative, the direction is to the left. If the bottom number is positive, the direction is upwards.  If the bottom number is negative, the direction is downwards.

  • Column vectors notation

Column vectors only have 2 numbers within the brackets; a top number and a bottom number. There is no need for any other punctuation marks such as commas or semicolons and there is no need for a line to separate the numbers.

Practice column vector questions

1. Write this vector as a column vector:

GCSE Quiz False

Draw a horizontal line and a vertical line and count the squares.

2. Write this vector as a column vector:

3. Write the vector \textbf{x} as a column vector:

The column vector for \textbf{x} is

4. Draw the vector

The top number of the column vector is 1 . This is the horizontal component. Use this to draw a horizontal line to the right. The bottom number of the column vector is 4 . This is the vertical component. Use this to draw a vertical line upwards.

5. Draw the vector

The top number of the column vector is -3 . This is the horizontal component. Use this to draw a horizontal line to the left. The bottom number of the column vector is -2 . This is the vertical component. Use this to draw a vertical line downwards.

6. Draw the vector

The top number of the column vector is -5 . This is the horizontal component. Use this to draw a horizontal line to the left. The bottom number of the column vector is 2 . This is the vertical component. Use this to draw a vertical line upwards.

Column vector GCSE questions

1.  Which is the correct column vector for this vector?

2.  The column vector \begin{pmatrix} \; 4 \;\\ \; a \; \end{pmatrix} represents:

What is the value of a ?

3. Write the column vector for this vector

(For the correct horizontal component)

(For the correct vertical component)

Learning checklist

You have now learned how to:

  • How to write a vector as a column vector
  •  How to draw a diagram of a column vector

The next lessons are

  • Loci and construction
  • Transformations
  • Circle theorems

Did you know?

Not covered in GCSE: we can transpose a column vector to write it as a row vector (and vice versa). These look like co-ordinates, but do not have commas.

Vectors can also be extended into A Level Maths and Further Maths by learning how to multiply two vectors together using the dot product. 

Column vectors are a simple example of matrices.  In GCSE maths we have a single column. Matrices are studied in A Level Further Maths. The number of columns and rows will be more than 1 . Matrix multiplication can be studied along with finding the inverse of a matrix. We can also find the determinant of a matrix and go further and look into eigenvalues and eigenvectors.

Still stuck?

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-- Vectors - PureMaths 3 - RP AS & A Level Mathematics

Topic outline.

  • Pure Mathematics 3 3.7 Vectors

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Lesson 1: Vector notation and basics

Teacher tutorial:.

how to write vectors on paper

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Lesson 2: Vector equations of lines

how to write vectors on paper

Lesson 3: Problem solving with vectors and lines

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SCI Journal

How to write a vector in LaTeX?

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vector in LaTeX

This article aims to show you a simple way to create a vector in LaTeX .

Regardless of where you look in science books, you will find at least a vector somewhere in one of its many representations, and that’s because it is used for many things in science subjects. And if you are using LaTeX you will want to know how to insert vectors in your document.

Table of Contents

According to its definition a vector is a geometric object that has magnitude and direction, and has properties according to vector algebra. Usually represented graphically as an arrow connecting two points A, B and denoted by

vector in LaTeX

Vec command

The standard command to call the arrow above the letter is \ vec {} it takes one value as an argument, but it has a disadvantage, it does not cover the entire argument. Instead you can use \ overrightarrow {} likewise it takes one value as argument, for example

vector in LaTeX

Vectors in Math mode

You probably already noticed it, to call the vector commands you have to use math mode, more examples

vector in LaTeX

Vector with physics package

Now the physics package allows you to write the same vector arrow above the letters but with more options, for example without having to use a command for the boldness. The command for the vector is \ va {} and \ va* {} and for the bold font \ vb {} and \ vb* {}. For example

vector in LaTeX

Two things to notice, first the commands with the * generated a cursive font, and second the \ vb {} and \ vb* {} command does not generate an arrow above the argument, this form is another way to represent vectors in textbooks and documents, especially in physics.

Now you have the knowledge on how to write vectors in LaTeX in multiple ways, I prefer the physics package to write it due to its smoothness and simplicity, but that is up to you to decide.

I hope this post was helpful in you path using LaTeX, and as always keep writing in LaTeX.

Further Reading

LaTex Tutorial on Symbols

  • How To Create A Cross Product Symbol In LaTeX
  • How to create a hat symbol in LaTeX?
  • How to create a prime symbol in LaTeX?
  • How to create an absolute value symbol in LaTeX?
  • How to create an approximate symbol in LaTeX?
  • How to create an intersection symbol in LaTeX?
  • How to create the empty set symbol in LaTeX?
  • How to write a degree symbol in LaTeX?
  • How to write a dot product in LaTeX?
  • How to Write a Greater Than Symbol in LaTeX?
  • How to write a norm symbol in LaTeX?
  • How to write A Plus-Minus Symbol in LaTeX
  • How to write a proportional to symbol in LaTeX?
  • How to write a real number symbol in LaTeX?
  • How to write a tilde symbol in LaTeX?
  • How to write a union symbol in LaTeX?
  • How to write an infinity symbol in LaTeX?
  • How to write bold text in LaTeX?
  • How To Write Dots Symbols In LaTeX?
  • How to write the arrow symbols in LaTeX
  • How to write the Degree celsius symbol in LaTeX?
  • How to write the equal or not equal symbol in LaTeX?
  • How to write the Euro symbol in LaTeX?
  • How to write the floor symbol in LaTeX?
  • How to write the gradient operator symbol in LaTeX
  • How To Write The Greater Than Or Equal To Symbol In LaTeX?
  • How to write the integer number symbol in LaTeX?
  • How to write the less than symbol in LaTeX?
  • How to write the Natural numbers symbol in LaTeX?
  • How to write the parallel symbol in LaTeX?
  • How to write the percent symbol in LaTeX?
  • How to write the square root symbol in LaTeX?
  • How To Write The Symbol For A Subset In LaTeX?
  • How to write the symbol for therefore in LaTeX?
  • How to write with the mathbb in LaTeX?

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Critical Writing Program: Decision Making - Spring 2024: Researching the White Paper

  • Getting started
  • News and Opinion Sites
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  • Substantive News Sources
  • What to Do When You Are Stuck
  • Understanding a citation
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  • Chicago Manual of Style: Citing Images
  • Researching the Op-Ed
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Research the White Paper

Researching the White Paper:

The process of researching and composing a white paper shares some similarities with the kind of research and writing one does for a high school or college research paper. What’s important for writers of white papers to grasp, however, is how much this genre differs from a research paper.  First, the author of a white paper already recognizes that there is a problem to be solved, a decision to be made, and the job of the author is to provide readers with substantive information to help them make some kind of decision--which may include a decision to do more research because major gaps remain. 

Thus, a white paper author would not “brainstorm” a topic. Instead, the white paper author would get busy figuring out how the problem is defined by those who are experiencing it as a problem. Typically that research begins in popular culture--social media, surveys, interviews, newspapers. Once the author has a handle on how the problem is being defined and experienced, its history and its impact, what people in the trenches believe might be the best or worst ways of addressing it, the author then will turn to academic scholarship as well as “grey” literature (more about that later).  Unlike a school research paper, the author does not set out to argue for or against a particular position, and then devote the majority of effort to finding sources to support the selected position.  Instead, the author sets out in good faith to do as much fact-finding as possible, and thus research is likely to present multiple, conflicting, and overlapping perspectives. When people research out of a genuine desire to understand and solve a problem, they listen to every source that may offer helpful information. They will thus have to do much more analysis, synthesis, and sorting of that information, which will often not fall neatly into a “pro” or “con” camp:  Solution A may, for example, solve one part of the problem but exacerbate another part of the problem. Solution C may sound like what everyone wants, but what if it’s built on a set of data that have been criticized by another reliable source?  And so it goes. 

For example, if you are trying to write a white paper on the opioid crisis, you may focus on the value of  providing free, sterilized needles--which do indeed reduce disease, and also provide an opportunity for the health care provider distributing them to offer addiction treatment to the user. However, the free needles are sometimes discarded on the ground, posing a danger to others; or they may be shared; or they may encourage more drug usage. All of those things can be true at once; a reader will want to know about all of these considerations in order to make an informed decision. That is the challenging job of the white paper author.     
 The research you do for your white paper will require that you identify a specific problem, seek popular culture sources to help define the problem, its history, its significance and impact for people affected by it.  You will then delve into academic and grey literature to learn about the way scholars and others with professional expertise answer these same questions. In this way, you will create creating a layered, complex portrait that provides readers with a substantive exploration useful for deliberating and decision-making. You will also likely need to find or create images, including tables, figures, illustrations or photographs, and you will document all of your sources. 

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  • Last Updated: Feb 15, 2024 12:28 PM
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CBSE Class 10 Hindi Paper Analysis 2024: Exam Review, Student Feedback and Expert View

Cbse class 10 hindi paper analysis 2024: here, students can find the cbse class 10 hindi paper analysis along with students’ feedback and question paper difficulty check..

Atul Rawal

CBSE Class 10 Board Exam 2024: The Central Board of Secondary Education (CBSE) Higher Secondary Hindi exam is now over. Students have left the exam centres with the question paper in their hands. On Wednesday, February 21, 2024, the CBSE Class 10 Hindi (Course A and Course B) exam was conducted. The exam started at 10:30 a.m. and was completed at 1:30 p.m. Thus, it took a total of three hours to finish the paper. 

Difference between CBSE Class 10 Hindi Course A and Course B

Cbse class 10 hindi question paper format 2024.

We will start our analysis by discussing the question paper format or design. This will give an idea of the number of questions and their distribution in the Class 10 Hindi paper, CBSE 2024. 

CBSE Class 10 Hindi Section-Wise Analysis

  • The questions in Section A were objective types, or MCQs. 
  • There were 10 comprehensive passages and poems that helped to find the answers.
  • Overall, the section was a bit twisted, but easy to understand.
  • This section comprises descriptive questions.
  • This section consumed a lot of time, so students had to rush in the end.
  • Overall, this section was moderate.

CBSE Class 10 Hindi Exam 2024 Difficulty Level: Student Reaction

Here we will analyse the difficulty level of the question paper based on the students' reactions after the exam's completion. The thing to highlight here is that only a few students were able to interact, so the verdict might vary with a higher percentage. 

CBSE Class 10 Hindi Exam 2024: Expert Opinion

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IMAGES

  1. How to Represent Vectors write Unit Vector and Graph them

    how to write vectors on paper

  2. How to write a vector in terms of two other vectors

    how to write vectors on paper

  3. How to write a vector in terms of two other vectors

    how to write vectors on paper

  4. Vector Notation

    how to write vectors on paper

  5. Writing a Single Column Vector

    how to write vectors on paper

  6. Representation of Vectors How to write a vector?

    how to write vectors on paper

VIDEO

  1. Vectors

  2. VECTORS SURE QUESTIONS (BASED ON MODEL QUESTION PATERN)

  3. Vectors-Practice questions

  4. Question Solve [Weekly 01]

  5. Vectors Past Paper Example 1

  6. Vectors Tutorial

COMMENTS

  1. Vectors and notation (article)

    There are lots of ways to write vectors. Here are the three we'll use most in this course. The little arrow on top of v → is a convention that indicates that v → refers to a vector. v → = ( 1, 2, 3) = [ 1 2 3] = 1 ı ^ + 2 ȷ ^ + 3 k ^ The first notation is what we discussed earlier.

  2. convention of writing vector symbols

    1 I am always confused when I am denoting vectors. From Wikipedia, it says the common convention of vectors are always in lowercase boldface letters or with arrow. However, our math teacher writes vector simply by lowercase letter (like u or v). For instance, u = i + j + k u = i + j + k. Is it possible? Thanks! vectors Share Cite Follow

  3. Vector Notation

    We can write vectors in several ways: Using an arrow, Using boldface Underlined. E.g. \overrightarrow {AB} =\textbf {a}=\underline {a} AB = a = a This diagram shows a vector representing the move from point A to point B. Parallel vectors with the same direction and the same length are the same (equivalent): E.g.

  4. Vector notation

    Using the modern terms cross product (×) and dot product (.), the quaternion product of two vectors p and q can be written pq = - p. q + p × q. In 1878, W. K. Clifford severed the two products to make the quaternion operation useful for students in his textbook Elements of Dynamic.

  5. 11.1: Vectors in the Plane

    A vector in a plane is represented by a directed line segment (an arrow). The endpoints of the segment are called the initial point and the terminal point of the vector. An arrow from the initial point to the terminal point indicates the direction of the vector. The length of the line segment represents its magnitude.

  6. PDF Introduction to vectors

    1. Introduction Vector quantities are extremely useful in physics. The important characteristic of a vector quan-tity is that it has both a magnitude (or size) and a direction. Both of these properties must be given in order to specify a vector completely. An example of a vector quantity is a displacement.

  7. vectors

    Using gridded paper, draw each of the following vectors: the vector a = (5 2) a → = ( 5 2) the vector b = (−3 4) b → = ( − 3 4) the vector c = ( 6 −3) c → = ( 6 − 3) the vector d = (0 5) d → = ( 0 5) the vector e = (−2 −5) e → = ( − 2 − 5) the vector f = (−4 0) f → = ( − 4 0) the vector g = (1 6) g → = ( 1 6) the vector h = (5 5) h → = ( 5 5)

  8. How to write a vector in three different forms

    I make short, to-the-point online math tutorials. I struggled with math growing up and have been able to use those experiences to help students improve in ma...

  9. What is the correct way of writing a vector element?

    What is the correct way of writing a vector element? Ask Question Asked 8 years, 9 months ago Modified 2 years, 2 months ago Viewed 2k times 4 When using the convention making a label bold to indicate a vector, should you still use the bold if you are only referring to a single element of the vector?

  10. 5.1 Vector Addition and Subtraction: Graphical Methods

    The head-to-tail method is a graphical way to add vectors. The tail of the vector is the starting point of the vector, and the head (or tip) of a vector is the pointed end of the arrow. The following steps describe how to use the head-to-tail method for graphical vector addition. Let the x -axis represent the east-west direction.

  11. A.1: Vectors, Mappings, and Matrices

    As vectors are arrows, when we want to give a name to a vector, we draw a little arrow above it: \[\vec{x} \nonumber \] Another popular notation is \(\mathbf{x}\), although we will use the little arrows. It may be easy to write a bold letter in a book, but it is not so easy to write it by hand on paper or on the board.

  12. 2.2: Vector Equations and Spans

    Definition 2.2. 1: Vector Equation. A vector equation is an equation involving a linear combination of vectors with possibly unknown coefficients. Note 2.2. 1. Asking whether or not a vector equation has a solution is the same as asking if a given vector is a linear combination of some other given vectors.

  13. How do you represent vectors?

    To really use these, you need unit vectors. This is what they look like: The little ^ over the x means that it is a unit vector. A unit vector is a vector that has a magnitude of 1 with no units ...

  14. Vectors

    direction and magnitude (size). A scalar quantity has only magnitude. A vector can be represented by a line segment labelled with an arrow. A vector between two points A and B is described as: \...

  15. Vectors

    1. Vector notation Vectors can be represented by a straight line segment with an arrow to show the direction of the vector (a directed line segment). These are also known as Euclidean vectors. This diagram shows a vector representing the move from point A to point B.

  16. Vectors

    The most common way is to first break up vectors into x and y parts, like this: The vector a is broken up into the two vectors a x and a y (We see later how to do this.) Adding Vectors. We can then add vectors by adding the x parts and adding the y parts: The vector (8, 13) and the vector (26, 7) add up to the vector (34, 20)

  17. Column Vector

    In order to write a vector as a column vector: Work out the horizontal component ( \textbf {x} x component). Work out the vertical component ( \textbf {y} y component). Write the column vector. How to write a column vector Column vector worksheet Get your free column vector worksheet of 20+ questions and answers.

  18. How do you write a vector clearly on one line?

    I've learned to write a vector on one line as for example $(a,b,c)$. This can get very confusing if each component of the vector is an expression of several terms like this: $(a+b,b+c,a+b)$ since the commas get mixed up in the mess of signs.

  19. -- Vectors

    Lesson 3: Problem solving with vectors and lines. Lesson 3 worksheet F. Lesson 3 - Making Quadrilaterals.

  20. How to write a vector in LaTeX? 2024

    Now the physics package allows you to write the same vector arrow above the letters but with more options, for example without having to use a command for the boldness. The command for the vector is \ va {} and \ va* {} and for the bold font \ vb {} and \ vb* {}. For example. 1. 2.

  21. Vectors

    To use this vector calculator simply enter the x and y value of your two vectors below. Make sure to separate the x and y value with a comma. I put an example below so you can see how it is done. 1. Next to add/subtract/dot product/find the magnitude simply press the empty white circle next to the "ADDITION" if you want to add the vectors and ...

  22. How does one write boldface characters on paper or a chalkboard?

    $\begingroup$ When I write vectors by hand (and only when I need to distinguish them from scalars) I fall back to the old-school $\vec c$ notation. $\endgroup$ - user856 Sep 16, 2016 at 19:23

  23. Researching the White Paper

    For example, if you are trying to write a white paper on the opioid crisis, you may focus on the value of providing free, sterilized needles--which do indeed reduce disease, and also provide an opportunity for the health care provider distributing them to offer addiction treatment to the user. However, the free needles are sometimes discarded ...

  24. CBSE Class 10 Hindi Paper Analysis 2024: Exam Review, Student Feedback

    Students have left the exam centres with the question paper in their hands. Today, on Wednesday, February 21, 2024, the CBSE Class 10 Hindi (Course A and Course B) exam was conducted. The exam ...