vector addition assignment

5.1 Vector Addition and Subtraction: Graphical Methods

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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Linear algebra

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1.6: Vector Addition

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Adding Vectors in Two Dimensions

In the following image, vectors A and B represent the two displacements of a person who walked 90. m east and then 50. m north. We want to add these two vectors to get the vector sum of the two movements.

The graphical process for adding vectors in two dimensions is to place the tail of the second vector on the arrow head of the first vector as shown above.

The sum of the two vectors is the vector that begins at the origin of the first vector and goes to the ending of the second vector, as shown below.

If we are using totally graphic means of adding these vectors, the magnitude of the sum can be determined by measuring the length of the sum vector and comparing it to the original standard. We then use a compass to measure the angle of the summation vector.

If we are using calculation, we first determine the inverse tangent of 50 units divided by 90 units and get the angle of 29° north of east. The length of the sum vector can then be determined mathematically by the Pythagorean theorem, a2+b2=c2. In this case, the length of the hypotenuse would be the square root of (8100 + 2500), or 103 units.

If three or four vectors are to be added by graphical means, we would continue to place each new vector head to toe with the vectors to be added until all the vectors were in the coordinate system. The resultant, or sum, vector would be the vector from the origin of the first vector to the arrowhead of the last vector; the magnitude and direction of this sum vector would then be measured.

Mathematical Methods of Vector Addition

We can add vectors mathematically using trig functions, the law of cosines, or the Pythagorean theorem.

If the vectors to be added are at right angles to each other, such as the example above, we would assign them to the sides of a right triangle and calculate the sum as the hypotenuse of the right triangle. We would also calculate the direction of the sum vector by using an inverse sin or some other trig function.

Suppose, however, that we wish to add two vectors that are not at right angles to each other. Let’s consider the vectors in the following images.

The two vectors we are to add are a force of 65 N at 30° north of east and a force of 35 N at 60° north of west.

We know that vectors in the same dimension can be added by regular arithmetic. Therefore, we can resolve each of these vectors into components that lay on the axes as pictured below. The resolution of vectors reduces each vector to a component on the north-south axis and a component on the east-west axis.

After resolving each vector into two components, we can now mathematically determine the magnitude of the components. Once we have done that, we can add the components in the same direction arithmetically. This will give us two vectors that are perpendicular to each other and can be the legs of a right triangle.

The east-west component of the first vector is (65 N)(cos 30°) = (65 N)(0.866) = 56.3 N north

The north-south component of the first vector is (65 N)(sin 30°) = (65 N)(0.500) = 32.5 N north

The east-west component of the second vector is (35 N)(cos 60°) = (35 N)(0.500) = 17.5 N west

The north-south component of the second vector is (35 N)(sin 60°) = (35 N)(0.866) = 30.3 N north

The sum of the two east-west components is 56.3 N - 17.5 N = 38.8 N east

The sum of the two north-south components is 32.5 N + 30.3 N = 62.8 N north

We can now consider those two vectors to be the sides of a right triangle and find the length and direction of the hypotenuse using the Pythagorean Theorem and trig functions.

c=38.82+62.82=74 N

sin⁡ x=62.874 so x=sin−1⁡0.84 so x=58∘

The direction of the sum vector is 74 N at 58° north of east.

Perpendicular vectors have no components in the other direction. For example, if a boat is floating down a river due south, and you are paddling the boat due east, the eastward vector has no component in the north-south direction and therefore, has no effect on the north-south motion. If the boat is floating down the river at 5 mph south and you paddle the boat eastward at 5 mph, the boat continues to float southward at 5 mph. The eastward motion has absolutely no effect on the southward motion. Perpendicular vectors have NO effect on each other.

A motorboat heads due east at 16 m/s across a river that flows due north at 9.0 m/s.

Example 1.6.1

What is the resultant velocity of the boat?

Since the two motions are perpendicular to each other, they can be assigned to the legs of a right triangle and the hypotenuse (resultant) calculated.

c=a2+b2=(16 m/s)2+(9.0 m/s)2=18 m/s

sin⁡θ=9.018=0.500 and therefore θ=30∘

The resultant is 18 m/s at 30° north of east.

Example 1.6.2

If the river is 135 m wide, how long does it take the boat to reach the other side?

The boat is traveling across the river at 16 m/s due to the motor. The current is perpendicular and therefore has no effect on the speed across the river. The time required for the trip can be determined by dividing the distance by the velocity.

t=dv=135 m16 m/s=8.4 s

Example 1.6.3

The boat is traveling across the river for 8.4 seconds and therefore, it is also traveling downstream for 8.4 seconds. We can determine the distance downstream the boat will travel by multiplying the speed downstream by the time of the trip.

d downstream =(v downstream )(t)=(9.0 m/s)(8.4 s)=76 m

Use this PLIX Interactive to visualize how any vector can be broken down into separate x and y components:

Interactive Element

• Vectors can be added mathematically using geometry and trigonometry.
• Vectors that are perpendicular to each other have no effect on each other.
• Vector addition can be accomplished by resolving the vectors to be added into components those vectors, and then completing the addition with the perpendicular components.
• What is the total distance walked by the hiker?
• What is the displacement (on a straight line) of the hiker from the camp?
• While flying due east at 33 m/s, an airplane is also being carried due north at 12 m/s by the wind. What is the plane’s resultant velocity?
• Two students push a heavy crate across the floor. John pushes with a force of 185 N due east and Joan pushes with a force of 165 N at 30° north of east. What is the resultant force on the crate?
• An airplane flying due north at 90. km/h is being blown due west at 50. km/h. What is the resultant velocity of the plane?

Explore More

Use this resource to answer the questions that follow.

• What are the steps the teacher undertakes in order to calculate the resultant vector in this problem?
• How does she find the components of the individual vectors?
• How does she use the individual vector’s components to find the components of the resultant vector?
• Once the components are determined, how does she find the overall resultant vector?

Additional Resources

Real World Application: Banked With No Friction

Vector Addition

Vector addition finds its application in physical quantities where vectors are used to represent velocity, displacement, and acceleration.

• Adding the vectors geometrically is putting their tails together and thereby constructing a parallelogram. The sum of the vectors is the diagonal of the parallelogram that starts from the intersection of the tails.
• Adding vectors algebraically is adding their corresponding components.

In this article, let's learn about the addition of vectors, their properties, and various laws with solved examples.

What is the Vector Addition?

Vectors are represented as a combination of direction and magnitude and are written with an alphabet and an arrow over them (or) with an alphabet written in bold. Two vectors , a and b , can be added together using vector addition , and the resultant vector can be written as: a + b . Before learning about the properties of vector addition, we need to know about the conditions that are to be followed while adding vectors. The conditions are as follows:

• Vectors can be added only if they are of the same nature. For instance, acceleration should be added with only acceleration and not mass
• We cannot add vectors and scalars together

Consider two vectors C and D , where, C = C x i + C y j + C z k and D = D x i + D y j + Dzk. Then, the resultant vector (or vector sum formula) is R = C + D = (C x + D x )i + (C y + D y )j + (C z + C z ) k

Properties of Vector Addition

Vector addition is different from algebraic addition. Here are some of the important properties to be considered while doing vector addition:

Addition of Vectors Graphically

Vectors adding can be done using graphical and mathematical methods. These methods are as follows:

Vector Addition Using the Components

Triangle law of addition of vectors, parallelogram law of addition of vectors.

Vectors that are represented in cartesian coordinates can be decomposed into vertical and horizontal components. For instance, a vector A at an angle Φ, as shown in the below-given image, can be decomposed into its vertical and horizontal components as:

In the above image,

• A x , represents the component of vector A along the horizontal axis (x-axis), and
• A y , represents the component of vector A along the vertical axis (y-axis).

We can note that the three vectors form a right triangle and that the vector A can be expressed as:

A = A x + A y

Mathematically, using the magnitude and the angle of the given vector, we can determine the components of a vector.

A x = A cos Φ

A y = A sin Φ

For two vectors, if its horizontal and vertical components are given, then the resultant vector can be calculated. For instance, if the values of A x and A y are provided, then we will be able to calculate the angle and the magnitude of the vector A as follows:

| A | = √ (( A x ) 2 +( A y ) 2 )

And the angle can be found as:

Φ = tan-1 ( A y / A x )

Hence, we can conclude that:

• If the components of a vector are provided, then we can determine the resultant vector
• Likewise, we can determine the components of a vector using the above equations, if the vector is provided

Similarly, we can perform the addition on vectors using their components, if these vectors are expressed in ordered pairs i.e column vectors. For example, consider the two vectors P and Q .

P = (p 1 , p 2 )

Q = (q 1 , q 2 )

The resultant vector M can be obtained by performing vector addition on the two vectors P and Q , by adding the respective x and y components of these two vectors.

M = (p 1 +q 1 , p 2 + q 2 ).

This can be expressed explicitly as:

M x = p 1 + q 1

M y = p 2 + q 2 .

The magnitude formula to find the magnitude of the resultant vector M is: | M | = √ ((M x ) 2 +(M y ) 2 )

And the angle can be computed as Φ = tan-1 (M y / M x )

Laws of Vector Addition

There are two laws of vector addition (As mentioned in the previous section).

• Triangle law
• Parallelogram law

Using these two laws, we are going to prove that the sum of two vectors is obtained by attaching them head to tail and the vector sum is given by the vector that joins the free tail and free head. Let us study each of these laws in detail in the upcoming sections.

The famous triangle law can be used for the addition of vectors and this method is also called the head-to-tail method. As per this law, two vectors can be added together by placing them together in such a way that the first vector’s head joins the tail of the second vector. Thus, by joining the first vector’s tail to the head of the second vector, we can obtain the resultant vector sum. The addition of vectors using the triangle law can be with the following steps:

• First, the two vectors M and N are placed together in such a manner that the head of vector M connects the tail of vector N .
• And then, in order to find the sum, a resultant vector S is drawn in such a way that it connects the tail of M to the head of N .
• Thus, mathematically, the sum, or the resultant, vector S , in the below-given image can be expressed as S = M + N .

Thus, when the two vectors M and N are added using the triangle law, we can see that a triangle is formed by the two original vectors M and N , and the sum vector S .

Another law that can be used for the addition of vectors is the parallelogram law of the addition of vectors. Let’s take two vectors p and q , as shown below. They form the two adjacent sides of a parallelogram in their magnitude and direction. The sum p + q is represented in magnitude and direction by the diagonal of the parallelogram through their common point. This is the parallelogram law of vector addition.

In the above-given figure, using the Triangle law, we can conclude the following:

OP + PR = OR

OP + OQ = OR , since PR = OQ

Hence, we can conclude that the triangle laws of vector addition and the parallelogram law of vector addition are equivalent to each other.

Vector Addition Formulas

We use one of the following formulas to add two vectors a = <a 1 , a 2 , a 3 > and b = <b 1 , b 2 , b 3 >.

• If the vectors are in the component form then the vector sum formula is a + b = <a 1 + b 1 , a 2 + b 2 , a 3 + b 3 >.
• If the two vectors are arranged by attaching the head of one vector to the tail of the other, then their sum is the vector that joins the free head and free tail (by triangle law).
• If the two vectors represent the two adjacent sides of a parallelogram then the sum represents the diagonal vector that is drawn from the common point of both vectors (by parallelogram law).

Important Notes on Vector Addition:

Here is a list of a few points that should be remembered while studying the addition of vectors:

• Vectors are represented as a combination of direction and magnitude and they are drawn with an arrow representation.
• If the components of a vector are provided, then we can determine the resultant vector.
• The famous triangle law can be used for the addition of vectors and this method is also called the head-to-tail method.

☛ Related Articles:

Check out the following pages related to the addition of vectors:

• Adding Vectors Calculator
• Angle Between Two Vectors Calculator
• Triangle Inequality in Vectors

Examples of Addition of Vectors

Example 1: Find the addition of vectors PQ and QR , where PQ = (3, 4) and QR = (2, 6)

We will perform the vector addition by adding their corresponding components

PQ + QR = (3, 4) + (2, 6)

= (3 + 2, 4 + 6)

Answer: (5, 10).

Example 2: Two vectors are given along with their components: A = (2,3) and B = (2,-2). Calculate the magnitude and the angle of the sum C using their components.

Let us represent the components of the given vectors as:

• In the A , A x = 2 and A y = 3
• In the B , B x = 2 and B y = -2

Now, adding the two vectors,

A + B = (2, 3) + (2, -2) = (4, 1)

It can also be written as:

Here in C , C x = 4 and C y = 1

The magnitude of the resultant vector C can be calculated as:

| C | = √ ((C x ) 2 +(C y ) 2 )

| C | = √ ((4) 2 + (1) 2 )

= √ (16 + 1)

| C | = √ 17 = 4.123 units (Approximately)

And the angle can be calculated as follows:

Φ = tan-1 (C y / C x )

Φ = tan-1 (1/4)

Φ ≈ 14.04 degrees

Answer: Thus, the magnitude of the resultant vector | C | = 4.123 units (Approximately) and the angle Φ = 14.04 degrees

Example 3: If a = <1, -1> and b = <2, 1> then find the unit vector in the direction of addition of vectors a and b .

The vector sum is:

a + b = <1, -1> + <2, 1> = <1 + 2, -1 + 1> = <3, 0>

Its magnitude is, | a + b | = √(3 2 + 0 2 ) = √9 = 3.

The unit vector in the direction of vector addition is:

( a + b ) / | a + b | = <3, 0> / 3 = <1, 0>

Answer: The required unit vector is, <1, 0>.

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Practice Questions on Vector Addition

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FAQs on Vector Addition

What is the addition of vectors.

The addition of vectors means putting two or more vectors together. In the addition of vectors, we are adding two or more vectors using the addition operation in order to obtain a new vector that is equal to the sum of the two or more vectors. The sum of vectors a and b is written as a + b .

Example: Given two vectors, a = (2, 5) and b = (4, -2), the sum of vectors is (6,3)

What is the Formula For the Addition of Vectors?

This is the addition of vectors formula: Given two vectors a = (a 1 , a 2 ) and b = (b 1 , b 2 ), then the vector sum is, M = (a 1 + b 1 , a 2 + b 2 ) = (M x , M y ). In this case,

• magnitude of the resultant vector sum M = | M | = √ ((M x ) 2 +(M y ) 2 ) and
• the angle can be computed as θ = tan -1 (M y / M x )

What is the Vector Addition Rule?

To add two vectors that are in component form, we just add their corresponding components. To add two vectors geometrically, we use triangle law or parallelogram law.

What is the Formula of Parallelogram Law of Addition of Vectors?

As per the parallelogram law of addition of vectors , for two given vectors u and v enclosing an angle θ, the magnitude of the sum, | u + v |, is given by √( u 2 + v 2 +2 uv cos(θ)).

What is the Difference Between Vector addition and Subtraction?

Here are the differences between addition of vectors and the subtraction of vectors .

Is Addition of Vectors Commutative?

Yes, vectors adding is commutative ; for any two arbitrary vectors c , and d , c + d = d + c .

What is the Difference Between the Triangle Law of Vector Addition and the Parallelogram Law of Vector Addition?

For any two given vectors, as per the triangle law of vector addition, the third side of the triangle will become the resultant sum vector. Whereas, as per the parallelogram law of vector addition, the diagonal becomes the resultant sum vector.

What is the Associative Property of Addition of Vectors?

Addition is associative ; for any three arbitrary vectors a , b , and c , a + ( b + c ) = ( a + b ) + c. i.e, the order of addition does not matter.

What is the Triangle Law of Addition of Vectors?

The triangle law of the addition of vectors states that two vectors can be added together by placing them together in such a way that the first vector’s head joins the tail of the second vector. Thus, by joining the first vector’s tail to the head of the second vector, we can obtain the resultant sum vector.