how to solve a vector sum problem

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.

• 5.2 Vector Addition and Subtraction: Analytical 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
• 5.1 Vector Addition and Subtraction: Graphical 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:

• Define components of vectors
• Describe the analytical method of vector addition and subtraction
• Use the analytical method of vector addition and subtraction to solve problems

Teacher Support

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

• (F) express and interpret relationships symbolically in accordance with accepted theories to make predictions and solve problems mathematically, including problems requiring proportional reasoning and graphical vector addition
• (E) develop and interpret free-body force diagrams;
• (F) identify and describe motion relative to different frames of reference.

In addition, the High School Physics Laboratory Manual addresses content in this section in the lab titled: Motion in Two Dimensions, as well as the following standards:

• (F) express and interpret relationships symbolically in accordance with accepted theories to make predictions and solve problems mathematically, including problems requiring proportional reasoning and graphical vector addition.

Section Key Terms

Components of vectors.

For the analytical method of vector addition and subtraction, we use some simple geometry and trigonometry, instead of using a ruler and protractor as we did for graphical methods. However, the graphical method will still come in handy to visualize the problem by drawing vectors using the head-to-tail method. The analytical method is more accurate than the graphical method, which is limited by the precision of the drawing. For a refresher on the definitions of the sine, cosine, and tangent of an angle, see Figure 5.17 .

[BL] [OL] Review trigonometric concepts of sine, cosine, tangent and the Pythagorean theorem.

Since, by definition, cos θ = x / h cos θ = x / h , we can find the length x if we know h and θ θ by using x = h cos θ x = h cos θ . Similarly, we can find the length of y by using y = h sin θ y = h sin θ . These trigonometric relationships are useful for adding vectors.

When a vector acts in more than one dimension, it is useful to break it down into its x and y components. For a two-dimensional vector, a component is a piece of a vector that points in either the x- or y-direction. Every 2-d vector can be expressed as a sum of its x and y components.

For example, given a vector like A A in Figure 5.18 , we may want to find what two perpendicular vectors, A x A x and A y A y , add to produce it. In this example, A x A x and A y A y form a right triangle, meaning that the angle between them is 90 degrees. This is a common situation in physics and happens to be the least complicated situation trigonometrically.

A x A x and A y A y are defined to be the components of A A along the x - and y -axes. The three vectors, A A , A x A x , and A y A y , form a right triangle.

If the vector A A is known, then its magnitude A A (its length) and its angle θ θ (its direction) are known. To find A x A x and A y A y , its x - and y -components, we use the following relationships for a right triangle:

where A x A x is the magnitude of A in the x -direction, A y A y is the magnitude of A in the y -direction, and θ θ is the angle of the resultant with respect to the x -axis, as shown in Figure 5.19 .

[BL] [OL] [AL] Derive the formula for getting the magnitude and direction of a vector.

Misconception Alert

Students might be confused between the relationship A x   +   A y   =   A A x   +   A y   =   A , which shows the addition of vectors and A = A x 2 + A y 2 A = A x 2 + A y 2 which shows the addition of magnitudes of vectors.

Suppose, for example, that A A is the vector representing the total displacement of the person walking in a city, as illustrated in Figure 5.20 .

Then A = 10.3 blocks and θ = 29.1 ∘ θ = 29.1 ∘ , so that

This magnitude indicates that the walker has traveled 9 blocks to the east—in other words, a 9-block eastward displacement. Similarly,

indicating that the walker has traveled 5 blocks to the north—a 5-block northward displacement.

Analytical Method of Vector Addition and Subtraction

Calculating a resultant vector (or vector addition) is the reverse of breaking the resultant down into its components. If the perpendicular components A x A x and A y A y of a vector A A are known, then we can find A A analytically. How do we do this? Since, by definition,

we solve for θ θ to find the direction of the resultant.

Note that tan − 1 ( θ ) tan − 1 ( θ ) gives an angle in the first quadrant if A y / A x > 0 A y / A x > 0 and in the fourth quadrant if A y / A x < 0 A y / A x < 0 . If, in fact, both A x A x and A y A y are negative, or if A x A x is negative and A y A y positive, then θ θ , measured from the positive x x direction, is tan – 1 ( θ ) + 180 ° tan – 1 ( θ ) + 180 ° .

Since this is a right triangle, the Pythagorean theorem (x 2 + y 2 = h 2 ) for finding the hypotenuse applies. In this case, it becomes

Solving for A gives

In summary, to find the magnitude A A and direction θ θ of a vector from its perpendicular components A x A x and A y A y , as illustrated in Figure 5.21 , we use the following relationships:

[BL] [OL] [AL] Demonstrate a problem involving displacement by physically walking along the specified direction. Show how this can be represented on a graph. Explain that even when solving problems analytically; representing it on a graph would make it easier to visualize the problem.

Sometimes, the vectors added are not perfectly perpendicular to one another. An example of this is the case below, where the vectors A A and B B are added to produce the resultant R , R , as illustrated in Figure 5.22 .

If A A and B B represent two legs of a walk (two displacements), then R R is the total displacement. The person taking the walk ends up at the tip of R R . There are many ways to arrive at the same point. The person could have walked straight ahead first in the x -direction and then in the y -direction. Those paths are the x - and y -components of the resultant, R x R x and R y . R y . If we know R x R x and R y R y , we can find R R and θ θ using the equations R = R x 2 + R y 2 R = R x 2 + R y 2 and θ = t a n – 1 ( R y / R x ) θ = t a n – 1 ( R y / R x ) .

and find the y component of the resultant (as illustrated in Figure 5.24 ) by adding the y component of the vectors.

Now that we know the components of R , R , we can find its magnitude and direction.

• To get the magnitude of the resultant R, use the Pythagorean theorem. R = R x 2 + R y 2 R = R x 2 + R y 2
• To get the direction of the resultant θ = tan − 1 ( R y / R x ) . θ = tan − 1 ( R y / R x ) .

Watch Physics

Classifying vectors and quantities example.

This video contrasts and compares three vectors in terms of their magnitudes, positions, and directions.

• 0 units . All of them will cancel each other out.
• 5 units . Two of them will cancel each other out.
• 10 units . Two of them will add together to give the resultant.
• 15 units. All of them will add together to give the resultant.

Tips For Success

In the video, the vectors were represented with an arrow above them rather than in bold. This is a common notation in math classes.

Using the Analytical Method of Vector Addition and Subtraction to Solve Problems

Figure 5.25 uses the analytical method to add vectors.

Worked Example

Add the vector A A to the vector B B shown in Figure 5.25 , using the steps above. The x -axis is along the east–west direction, and the y -axis is along the north–south directions. A person first walks 53 .0 m 53 .0 m in a direction 20 .0° 20 .0° north of east, represented by vector A . A . The person then walks 34 .0 m 34 .0 m in a direction 63 .0 ° 63 .0 ° north of east, represented by vector B . B .

The components of A A and B B along the x - and y -axes represent walking due east and due north to get to the same ending point. We will solve for these components and then add them in the x-direction and y-direction to find the resultant.

First, we find the components of A A and B B along the x - and y -axes. From the problem, we know that A = 53.0  m, A = 53.0  m, θ A = 20.0 ∘, θ A = 20.0 ∘, B B = 34 .0 m 34 .0 m , and θ B = 63.0 ∘ θ B = 63.0 ∘ . We find the x -components by using A x = A cos θ A x = A cos θ , which gives

Similarly, the y -components are found using A y = A sin θ A A y = A sin θ A

The x - and y -components of the resultant are

Now we can find the magnitude of the resultant by using the Pythagorean theorem

Finally, we find the direction of the resultant

This example shows vector addition using the analytical method. Vector subtraction using the analytical method is very similar. It is just the addition of a negative vector. That is, A − B ≡ A + ( − B ) A − B ≡ A + ( − B ) . The components of – B B are the negatives of the components of B B . Therefore, the x - and y -components of the resultant A − B = R A − B = R are

and the rest of the method outlined above is identical to that for addition.

Practice Problems

What is the magnitude of a vector whose x -component is 4 cm and whose y -component is 3 cm?

What is the magnitude of a vector that makes an angle of 30° to the horizontal and whose x -component is 3 units?

Links To Physics

Atmospheric science.

Atmospheric science is a physical science , meaning that it is a science based heavily on physics. Atmospheric science includes meteorology (the study of weather) and climatology (the study of climate). Climate is basically the average weather over a longer time scale. Weather changes quickly over time, whereas the climate changes more gradually.

The movement of air, water and heat is vitally important to climatology and meteorology. Since motion is such a major factor in weather and climate, this field uses vectors for much of its math.

Vectors are used to represent currents in the ocean, wind velocity and forces acting on a parcel of air. You have probably seen a weather map using vectors to show the strength (magnitude) and direction of the wind.

Vectors used in atmospheric science are often three-dimensional. We won’t cover three-dimensional motion in this text, but to go from two-dimensions to three-dimensions, you simply add a third vector component. Three-dimensional motion is represented as a combination of x -, y - and z components, where z is the altitude.

Vector calculus combines vector math with calculus, and is often used to find the rates of change in temperature, pressure or wind speed over time or distance. This is useful information, since atmospheric motion is driven by changes in pressure or temperature. The greater the variation in pressure over a given distance, the stronger the wind to try to correct that imbalance. Cold air tends to be more dense and therefore has higher pressure than warm air. Higher pressure air rushes into a region of lower pressure and gets deflected by the spinning of the Earth, and friction slows the wind at Earth’s surface.

Finding how wind changes over distance and multiplying vectors lets meteorologists, like the one shown in Figure 5.26 , figure out how much rotation (spin) there is in the atmosphere at any given time and location. This is an important tool for tornado prediction. Conditions with greater rotation are more likely to produce tornadoes.

• Vectors have magnitude as well as direction and can be quickly solved through scalar algebraic operations.
• Vectors have magnitude but no direction, so it becomes easy to express physical quantities involved in the atmospheric science.
• Vectors can be solved very accurately through geometry, which helps to make better predictions in atmospheric science.
• Vectors have magnitude as well as direction and are used in equations that describe the three dimensional motion of the atmosphere.

Check Your Understanding

Between the analytical and graphical methods of vector additions, which is more accurate? Why?

• The analytical method is less accurate than the graphical method, because the former involves geometry and trigonometry.
• The analytical method is more accurate than the graphical method, because the latter involves some extensive calculations.
• The analytical method is less accurate than the graphical method, because the former includes drawing all figures to the right scale.
• The analytical method is more accurate than the graphical method, because the latter is limited by the precision of the drawing.

What is a component of a two dimensional vector?

• A component is a piece of a vector that points in either the x or y direction.
• A component is a piece of a vector that has half of the magnitude of the original vector.
• A component is a piece of a vector that points in the direction opposite to the original vector.
• A component is a piece of a vector that points in the same direction as original vector but with double of its magnitude.
• θ = cos − 1 ⁡ A y A x
• θ = cot − 1 ⁡ A y A x
• θ = sin − 1 ⁡ A y A x
• θ = tan − 1 ⁡ A y A x

How can we determine the magnitude of a vector if we know the magnitudes of its components?

• | A → | = A x + A y | A → | = A x + A y
• | A → | = A x 2 + A y 2 | A → | = A x 2 + A y 2
• | A → | = ( A x 2 + A y 2 ) 2 | A → | = ( A x 2 + A y 2 ) 2

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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Solving Problems with Vectors

We can use vectors to solve many problems involving physical quantities such as velocity, speed, weight, work and so on.

The velocity of moving object is modeled by a vector whose direction is the direction of motion and whose magnitude is the speed.

A ball is thrown with an initial velocity of 70 feet per second., at an angle of 35 ° with the horizontal. Find the vertical and horizontal components of the velocity.

Let v represent the velocity and use the given information to write v in unit vector form:

v   = 70 ( cos ( 35 ° ) ) i + 70 ( sin ( 35 ° ) ) j

Simplify the scalars, we get:

v   ≈ 57.34 i + 40.15 j

Since the scalars are the horizontal and vertical components of v ,

Therefore, the horizontal component is 57.34 feet per second and the vertical component is 40.15 feet per second.

Force is also represented by vector. If several forces are acting on an object, the resultant force experienced by the object is the vector sum of these forces.

Two forces F 1 and F 2 with magnitudes 20 and 30   lb , respectively, act on an object at a point P as shown. Find the resultant forces acting at P .

First we write F 1 and F 2 in component form:

v ≈ 57.34 i + 40.15 j

F 1 = ( 20 cos ( 45 ° ) ) i + ( 20 sin ( 45 ° ) ) j = 20 ( 2 2 ) i + 20 ( 2 2 ) j = 10 2 i + 10 2 j F 2 = ( 30 cos ( 150 ° ) ) i + ( 30 sin ( 150 ° ) ) j = 30 ( − 3 2 ) i + 30 ( 1 2 ) j = − 15 3 i + 15 j

So, the resultant force F is

F = F 1 + F 2 = ( 10 2   i + 10 2 j ) + ( − 15 3   i + 15 j ) = ( 10 2 − 15 3 ) i + ( 10 2 + 15 ) j ≈ − 12 i + 29 j

A force is given by the vector F = ⟨ 2 , 3 ⟩ and moves an object from the point ( 1 , 3 ) to the point ( 5 , 9 ) . Find the work done.

First we find the Displacement.

The displacement vector is

D = ⟨ 5 − 1 , 9 − 3 ⟩ = ⟨ 4 , 6 ⟩ .

By using the formula, the work done is

W = F ⋅ D = ⟨ 2 , 3 ⟩ ⋅ ⟨ 4 , 6 ⟩ = 26

If the unit of force is pounds and the distance is measured in feet, then the work done is 26 ft-lb.

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• \begin{pmatrix}1&0&3\end{pmatrix}+\begin{pmatrix}-1&4&2\end{pmatrix}
• (-3)\cdot \begin{pmatrix}1&5&0\end{pmatrix}
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• \begin{pmatrix}1&2&3\end{pmatrix}\times\begin{pmatrix}1&5&7\end{pmatrix}
• angle\:\begin{pmatrix}2&-4&-1\end{pmatrix},\:\begin{pmatrix}0&5&2\end{pmatrix}
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• What are vectors in math?
• In math, a vector is an object that has both a magnitude and a direction. Vectors are often represented by directed line segments, with an initial point and a terminal point. The length of the line segment represents the magnitude of the vector, and the arrowhead pointing in a specific direction represents the direction of the vector.
• What are the types of vectors?
• The common types of vectors are cartesian vectors, column vectors, row vectors, unit vectors, and position vectors.
• How do you add two vectors?
• To add two vectors, add the corresponding components from each vector. Example: the sum of (1,3) and (2,4) is (1+2,3+4), which is (3,7)

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• Advanced Math Solutions – Vector Calculator, Advanced Vectors In the last blog, we covered some of the simpler vector topics. This week, we will go into some of the heavier... Read More

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Weisstein, Eric W. "Vector Sum." From MathWorld --A Wolfram Web Resource. https://mathworld.wolfram.com/VectorSum.html

Subject classifications

• Scalars And Vectors
• Addition Of Vectors

Vector Addition and Subtraction

The vector addition is not as straightforward as the addition of scalars. Vectors have both magnitude and direction, and one cannot simply add two vectors to obtain their sum. To better understand this, let us consider an example of a car travelling 10 miles North and 10 miles South. Here, the total distance travelled is 20 miles, but the displacement is zero. The North and South displacements are each vector quantities, and the opposite directions cause the individual displacements to cancel each other out. In this article, let us explore ways to carry out vector addition and vector subtraction.

Vector Addition: Triangle, Parallelogram and Polygon Law of Vectors

As already discussed, vectors cannot be added algebraically. Following are a few points to remember while adding vectors:

• Vectors are added geometrically and not algebraically.
• Vectors whose resultant have to be calculated behave independently.
• Vector Addition is nothing but finding the resultant of a number of vectors acting on a body.
• Vector Addition is commutative. This means that the resultant vector is independent of the order of vectors. \(\begin{array}{l}\vec{A} + \vec{B} = \vec{B} + \vec{A}\end{array} \)

Triangle Law of Vector Addition

The vector addition is done based on the triangle law. Let us see what the triangle law of vector addition is:

Suppose there are two vectors, a and b.

Draw a line AB representing vector a with A as the tail and B as the head. Draw another line BC representing vector b with B as the tail and C as the head. Now join the line AC with A as the tail and C as the head. The line AC represents the resultant sum of the vectors a and b.

The line AC represents the resultant sum of the vectors a and b.

The magnitude of vectors a and b is:

a = magnitude of vector a

b = magnitude of vector b

θ = angle between vector a and b

Let the resultant make an angle of Φ with vector a, then:

Let us understand this using an example. Suppose two vectors have equal magnitude A, and they make an angle θ with each other. Now, to find the magnitude and direction of the resultant, we will use the formulas mentioned above.

Let the magnitude of the resultant vector be B.

Let’s say that the resultant vector makes an angle Ɵ with the first vector.

Read More: Triangle Law of Vector Addition

Parallelogram Law of Vector Addition

The vector addition may also be understood by the law of parallelogram. The law states, “If two vectors acting simultaneously at a point are represented in magnitude and direction by the two sides of a parallelogram drawn from a point, their resultant is given in magnitude and direction by the diagonal of the parallelogram passing through that point.”

According to this law, if two vectors, P and Q, are represented by two adjacent sides of a parallelogram pointing outwards, as shown in the figure below, then the diagonal drawn through the intersection of the two vectors represent the resultant.

In the Parallelogram Law of Vector Addition, the magnitude of the resultant is given by:

The direction of the resultant vector is determined as follows: \(\begin{array}{l}\tan\phi=\frac{CE}{AE}=\frac{Q\sin\theta}{P+Q\cos\theta}\end{array} \) \(\begin{array}{l}\theta=\tan^{-1}[\frac{Q\sin\theta}{P+Q\cos\theta}]\end{array} \)

Polygon Law of Vector

From the triangle law of vector addition, we know that triangle OLM can be expressed as vector OM is the resultant of the vectors OL and LM.

Again, applying the triangle law of vector addition to triangle OMN,

From eq.2, we get,

Considering vector ON=vector R, the equation becomes

Click the below video to understand the application of the polygon law of vector addition

Properties of Vector Addition

The vector addition is the sum of multiple (two or more) vectors. Two laws related to the addition of vectors are parallelogram law and triangle law. Similarly, the properties related to vector addition are:

• Associative Property
• Commutative Property

Vector Subtraction

The subtraction of two vectors is similar to addition. Suppose vector a is to be subtracted from vector b.

vector a – vector b can be said as the addition of vectors a and -b. Thus, the addition formula can be applied as:

vector (-b) is nothing but vector b reversed in direction.

Related articles:

Frequently Asked Questions – FAQs

What is the maximum and minimum sum of two vectors.

The maximum sum of two vectors is obtained when the two vectors are directed in the same direction. The minimum sum is obtained when the two vectors are directed in the opposite direction.

Is vector addition commutative?

Yes, vector addition is commutative.

Is vector addition applicable to any two vectors?

No, vector addition does not apply to any two vectors. Two vectors are added only when they are of the same type and nature. For instance, two velocity vectors can be added, but one velocity vector and one force vector cannot be added.

State the associative property of vector addition.

The associative property of vector addition states that the sum of the vectors remains the same regardless of the order in which they are arranged.

Can the sum of two vectors have a zero?

Yes, two vectors with equal magnitude and pointing in opposite directions will have the sum equal to zero.

Watch the video and learn about the analytical method of Vector Addition

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Vector operations

In this tutorial we'll learn how to find: magnitude , dot product , angle between two vectors and cross product of two vectors.

1 : Magnitude

Magnitude is the vector length. The formula for magnitude of a vector $ \vec{v} = (v_1, v_2) $ is:

Example 01: Find the magnitude of the vector $ \vec{v} = (4, 2) $.

In this example we have $ v_1 = 4 $ and $ v_2 = 2 $ so the magnitude is:

Example 02: Find the magnitude of the vector $ \vec{v} = \left(\dfrac{2}{3}, \sqrt{3}, 2\right) $.

2 : Dot product

The formula for the dot product of vectors $ \vec{v} = (v_1, v_2) $ and $ \vec{w} = (w_1, w_2) $ is

Two vectors are orthogonal to each other if their dot product is equal zero.

Example 03: Calculate the dot product of $ \vec{v} = \left(4, 1 \right) $ and $ \vec{w} = \left(-1, 5 \right) $. Check if the vectors are mutually orthogonal.

To find the dot product we use the component formula:

Since the dot product is not equal zero we can conclude that vectors ARE NOT orthogonal.

Example 04: Find the dot product of the vectors $ \vec{v_1} = \left(\dfrac{1}{2}, \sqrt{3}, 5 \right) $ and $ \vec{v_2} = \left( 4, -\sqrt{3}, 10 \right) $.

3 : Angle between two vectors

To find the angle $ \alpha $ between vectors $ \vec{a} $ and $ \vec{b} $, we use the following formula:

Note that $ \vec{a} \cdot \vec{b} $ is a dot product while $\|\vec{a}\|$ and $\|\vec{b}\|$ are magnitudes of vectors $ \vec{a} $ and $ \vec{b}$.

Example 05: Find the angle between vectors $ \vec{a} = ( 4, 3) $ and $ \vec{b} = (-2, 2) $.

First we will find the dot product and magnitudes:

Now we'll find the $ \cos \alpha $

The angle $ \alpha $ is:

Example 06: Find the angle between vectors $ \vec{v_1} = \left(2, 1, -4 \right) $ and $ \vec{v_2} = \left( 3, -5, 2 \right) $.

4 : Cross product

The cross product of vectors $ \vec{v} = (v_1,v_2,v_3) $ and $ \vec{w} = (w_1,w_2,w_3) $ is given by the formula:

Note that the cross product requires both of the vectors to be in three dimensions.

If the two vectors are parallel than the cross product is equal zero.

Example 07: Find the cross products of the vectors $ \vec{v} = ( -2, 3 , 1) $ and $ \vec{w} = (4, -6, -2) $. Check if the vectors are parallel.

We'll find cross product using above formula

Since the cross product is zero we conclude that the vectors are parallel.

Example 08: Find the cross products of the vectors $ \vec{v_1} = \left(4, 2, -\dfrac{3}{2} \right) $ and $ \vec{v_2} = \left(\dfrac{1}{2}, 0, 2 \right) $.

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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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Precalculus

Course: precalculus   >   unit 6.

• Vector word problem: resultant velocity
• Vector word problem: hiking
• Vector word problem: resultant force

Vector word problems

• Vectors: FAQ
• His first stop is 6  km ‍   to the east and 3  km ‍   to the south from his house.
• His second stop is 2  km ‍   to the west and 1  km ‍   to the south from the first stop.
• His third stop is 7  km ‍   to the west and 5  km ‍   to the north from the second stop.
• Your answer should be
• an integer, like 6 ‍  
• a simplified proper fraction, like 3 / 5 ‍  
• a simplified improper fraction, like 7 / 4 ‍  
• a mixed number, like 1   3 / 4 ‍  
• an exact decimal, like 0.75 ‍  
• a multiple of pi, like 12   pi ‍   or 2 / 3   pi ‍  

Engineering Statics: Open and Interactive

Daniel W. Baker, William Haynes

Section 2.6 Vector Addition

Key questions.

• How do you set up vectors for graphical addition using the Triangle Rule?
• Does it matter which vector you start with when using the Triangle Rule?
• Why can you separate a two-dimensional vector equation into two independent equations to solve for up to two unknowns?
• If you and another student define vectors using different direction coordinate systems, will you end up with the same resultant vector?
• What is the preferred technique to add vectors in three-dimensional systems?

Subsection 2.6.1 Triangle Rule of Vector Addition

• Triangle Rule. Place the tail of one vector at the tip of the other vector, then draw the resultant from the first vector’s tail to the final vector’s tip.
• Parallelogram Rule. Place both vectors’ tails at the origin, then complete a parallelogram with lines parallel to each vector through the tip of the other. The resultant is equal to the diagonal from the tails to the opposite corner.

Instructions .

Subsection 2.6.2 orthogonal components, subsection 2.6.3 graphical vector addition, subsection 2.6.4 trigonometric vector addition.

• drawing a quick diagram using either rule,
• identifying three known sides or angles,
• using trigonometry to solve for the unknown sides and angles.
• There are only three sides in a triangle; thus vectors can only be added two at a time. If you need to add three or more vectors using this method, you must add the first two, then add the third to that sum and so on.
• If you fail to draw the correct vector triangle or identify the known sides and angles, you will not find the correct answer.
• The trigonometric functions are scalar functions. They are quick ways of solving for the magnitudes of vectors and the angle between vectors, but you may still need to find the vector components from a given datum.

Subsection 2.6.5 Algebraic Addition of Components

• Find the scalar components of each component vector in the \(x\) and \(y\) directions using the P to R procedure described in Subsection 2.3.3 .
• Algebraically sum the scalar components in each coordinate direction. The scalar components will be positive if they point right or up, negative if they point left or down. These sums are the scalar components of the resultant.
• Resolve the resultant’s components to find the magnitude and direction of the resultant vector using the R to P procedure described in Subsection 2.3.3 .

Example 2.6.4 . Vector Addition.

Subsection 2.6.6 vector subtraction.

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This web page is designed to provide some additional practice with the use of scaled vector diagrams for the addition of two or more vectors. Your time will be best spent if you read each practice problem carefully, attempt to solve the problem with a scaled vector diagram, and then check your answer. You are cautioned to avoid making a quick reference to the solution prior to making your own attempt at the solution. Such a habit is likely to fail at nurturing the ability to draw a scaled vector addition diagram. If the solution to these practice problems are still not meaningful, you are encouraged to obtain some on-line help in The Physics Classroom.  Visit the page on vector addition .

NOTE: Since your answers were determined using a scaled vector diagram, small errors in the measurement of the direction and magnitude of any one of the vectors may lead to small differences between your answers and the correct ones which are shown here. Do not  have a cow.

1.  Add the following vectors and determine the resultant.

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Definition of vector sum

Examples of vector sum in a sentence.

These examples are programmatically compiled from various online sources to illustrate current usage of the word 'vector sum.' Any opinions expressed in the examples do not represent those of Merriam-Webster or its editors. Send us feedback about these examples.

Word History

circa 1890, in the meaning defined above

Dictionary Entries Near vector sum

vector space

Cite this Entry

“Vector sum.” Merriam-Webster.com Dictionary , Merriam-Webster, https://www.merriam-webster.com/dictionary/vector%20sum. Accessed 22 Nov. 2023.

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Civil 3D Tips, Tricks, and Unnatural Acts: AU 2023 Edition

Description.

This session will be dedicated entirely to productivity techniques that will help improve your daily operations when working with Civil 3D software. These tips include automating regular tasks, using standard functions in new ways, exploring underutilized features, and even exploiting a handful of undocumented commands. We'll present the information using a real-world problem-solving context, rather than simply going feature by feature, so you can fully appreciate the "why" in addition to the "how." Come join us for this 90-minute session as we show you numerous ways to help you get the most from your Civil 3D investment.

Key Learnings

• Learn how to automate several tasks.
• Learn how to make more use of underused features.
• Learn how to use several standard functions in new ways.
• Discover several useful undocumented commands.