Vectors
Learning Objectives
By the end of this section, you will be able to:- View vectors geometrically.
- Find magnitude and direction.
- Perform vector addition and scalar multiplication.
- Find the component form of a vector.
- Find the unit vector in the direction of .
- Perform operations with vectors in terms of and .
- Find the dot product of two vectors.

A Geometric View of Vectors
A vector is a specific quantity drawn as a line segment with an arrowhead at one end. It has an initial point, where it begins, and a terminal point, where it ends. A vector is defined by its magnitude, or the length of the line, and its direction, indicated by an arrowhead at the terminal point. Thus, a vector is a directed line segment. There are various symbols that distinguish vectors from other quantities:- Lower case, boldfaced type, with or without an arrow on top such as .
- Given initial point and terminal point , a vector can be represented as . The arrowhead on top is what indicates that it is not just a line, but a directed line segment.
- Given an initial point of and terminal point , a vector may be represented as .
In Figure 2, we see the original vector and the position vector .

A General Note: Properties of Vectors
A vector is a directed line segment with an initial point and a terminal point. Vectors are identified by magnitude, or the length of the line, and direction, represented by the arrowhead pointing toward the terminal point. The position vector has an initial point at and is identified by its terminal point .Example 1: Find the Position Vector
Consider the vector whose initial point is and terminal point is . Find the position vector.Solution
The position vector is found by subtracting one x-coordinate from the other x-coordinate, and one y-coordinate from the other y-coordinate. Thus
The position vector begins at and terminates at . The graphs of both vectors are shown in Figure 3.

Example 2: Drawing a Vector with the Given Criteria and Its Equivalent Position Vector
Find the position vector given that vector has an initial point at and a terminal point at , then graph both vectors in the same plane.Solution
The position vector is found using the following calculation:
Thus, the position vector begins at and terminates at .

Finding Magnitude and Direction
To work with a vector, we need to be able to find its magnitude and its direction. We find its magnitude using the Pythagorean Theorem or the distance formula, and we find its direction using the inverse tangent function.A General Note: Magnitude and Direction of a Vector
Given a position vector , the magnitude is found by . The direction is equal to the angle formed with the x-axis, or with the y-axis, depending on the application. For a position vector, the direction is found by , as illustrated in Figure 5.
Example 3: Finding the Magnitude and Direction of a Vector
Find the magnitude and direction of the vector with initial point and terminal point . Draw the vector.Solution
First, find the position vector.
We use the Pythagorean Theorem to find the magnitude.
The direction is given as
However, the angle terminates in the fourth quadrant, so we add 360° to obtain a positive angle. Thus, .

Example 4: Showing That Two Vectors Are Equal
Show that vector v with initial point at and terminal point at is equal to vector u with initial point at and terminal point at . Draw the position vector on the same grid as v and u. Next, find the magnitude and direction of each vector.Solution
As shown in Figure 7, draw the vector starting at initial and terminal point . Draw the vector with initial point and terminal point . Find the standard position for each. Next, find and sketch the position vector for v and u. We have
Since the position vectors are the same, v and u are the same.
An alternative way to check for vector equality is to show that the magnitude and direction are the same for both vectors. To show that the magnitudes are equal, use the Pythagorean Theorem.
As the magnitudes are equal, we now need to verify the direction. Using the tangent function with the position vector gives
However, we can see that the position vector terminates in the second quadrant, so we add . Thus, the direction is .

Performing Vector Addition and Scalar Multiplication
Now that we understand the properties of vectors, we can perform operations involving them. While it is convenient to think of the vector as an arrow or directed line segment from the origin to the point , vectors can be situated anywhere in the plane. The sum of two vectors u and v, or vector addition, produces a third vector u+ v, the resultant vector. To find u + v, we first draw the vector u, and from the terminal end of u, we drawn the vector v. In other words, we have the initial point of v meet the terminal end of u. This position corresponds to the notion that we move along the first vector and then, from its terminal point, we move along the second vector. The sum u + v is the resultant vector because it results from addition or subtraction of two vectors. The resultant vector travels directly from the beginning of u to the end of v in a straight path, as shown in Figure 8.

Example 5: Adding and Subtracting Vectors
Given and , find two new vectors u + v, and u − v.Solution
To find the sum of two vectors, we add the components. Thus,
See Figure 10(a).
To find the difference of two vectors, add the negative components of to . Thus,
See Figure 10(b).

Multiplying By a Scalar
While adding and subtracting vectors gives us a new vector with a different magnitude and direction, the process of multiplying a vector by a scalar, a constant, changes only the magnitude of the vector or the length of the line. Scalar multiplication has no effect on the direction unless the scalar is negative, in which case the direction of the resulting vector is opposite the direction of the original vector.A General Note: Scalar Multiplication
Scalar multiplication involves the product of a vector and a scalar. Each component of the vector is multiplied by the scalar. Thus, to multiply by , we have
Only the magnitude changes, unless is negative, and then the vector reverses direction.
Example 6: Performing Scalar Multiplication
Given vector , find 3v, , and −v.Solution
See Figure 11 for a geometric interpretation. If , then
Analysis of the Solution
Notice that the vector 3v is three times the length of v, is half the length of v, and –v is the same length of v, but in the opposite direction.Example 7: Using Vector Addition and Scalar Multiplication to Find a New Vector
Given and , find a new vector w = 3u + 2v.Solution
First, we must multiply each vector by the scalar.
Then, add the two together.
So, .
Finding Component Form
In some applications involving vectors, it is helpful for us to be able to break a vector down into its components. Vectors are comprised of two components: the horizontal component is the direction, and the vertical component is the direction. For example, we can see in the graph in Figure 12 that the position vector comes from adding the vectors v1 and v2. We have v1 with initial point and terminal point .We also have v2 with initial point and terminal point .
Therefore, the position vector is
Using the Pythagorean Theorem, the magnitude of v1 is 2, and the magnitude of v2 is 3. To find the magnitude of v, use the formula with the position vector.
The magnitude of v is . To find the direction, we use the tangent function .

Example 8: Finding the Components of the Vector
Find the components of the vector with initial point and terminal point .Solution
First find the standard position.
See the illustration in Figure 13.

Finding the Unit Vector in the Direction of v
In addition to finding a vector’s components, it is also useful in solving problems to find a vector in the same direction as the given vector, but of magnitude 1. We call a vector with a magnitude of 1 a unit vector. We can then preserve the direction of the original vector while simplifying calculations. Unit vectors are defined in terms of components. The horizontal unit vector is written as and is directed along the positive horizontal axis. The vertical unit vector is written as and is directed along the positive vertical axis.
A General Note: The Unit Vectors
If is a nonzero vector, then is a unit vector in the direction of . Any vector divided by its magnitude is a unit vector. Notice that magnitude is always a scalar, and dividing by a scalar is the same as multiplying by the reciprocal of the scalar.Example 9: Finding the Unit Vector in the Direction of v
Find a unit vector in the same direction as .Solution
First, we will find the magnitude.
Then we divide each component by , which gives a unit vector in the same direction as v:
or, in component form

The vector i j is the unit vector in the same direction as v .
Performing Operations with Vectors in Terms of i and j
So far, we have investigated the basics of vectors: magnitude and direction, vector addition and subtraction, scalar multiplication, the components of vectors, and the representation of vectors geometrically. Now that we are familiar with the general strategies used in working with vectors, we will represent vectors in rectangular coordinates in terms of i and j.A General Note: Vectors in the Rectangular Plane
Given a vector with initial point and terminal point , v is written as
The position vector from to , where and , is written as v = ai + bj. This vector sum is called a linear combination of the vectors i and j.
The magnitude of v = ai + bj is given as .

Example 10: Writing a Vector in Terms of i and j
Given a vector with initial point and terminal point , write the vector in terms of and .Solution
Begin by writing the general form of the vector. Then replace the coordinates with the given values.Example 11: Writing a Vector in Terms of i and j Using Initial and Terminal Points
Given initial point and terminal point , write the vector in terms of and .Solution
Begin by writing the general form of the vector. Then replace the coordinates with the given values.Performing Operations on Vectors in Terms of i and j
When vectors are written in terms of and , we can carry out addition, subtraction, and scalar multiplication by performing operations on corresponding components.A General Note: Adding and Subtracting Vectors in Rectangular Coordinates
Given v = ai + bj and u = ci + dj, thenExample 12: Finding the Sum of the Vectors
Find the sum of and .Solution
According to the formula, we haveCalculating the Component Form of a Vector: Direction
We have seen how to draw vectors according to their initial and terminal points and how to find the position vector. We have also examined notation for vectors drawn specifically in the Cartesian coordinate plane using . For any of these vectors, we can calculate the magnitude. Now, we want to combine the key points, and look further at the ideas of magnitude and direction. Calculating direction follows the same straightforward process we used for polar coordinates. We find the direction of the vector by finding the angle to the horizontal. We do this by using the basic trigonometric identities, but with replacing .A General Note: Vector Components in Terms of Magnitude and Direction
Given a position vector and a direction angle ,
Thus, , and magnitude is expressed as .
Example 13: Writing a Vector in Terms of Magnitude and Direction
Write a vector with length 7 at an angle of 135° to the positive x-axis in terms of magnitude and direction.Solution
Using the conversion formulas and , we find that
This vector can be written as or simplified as
Try It 4
A vector travels from the origin to the point . Write the vector in terms of magnitude and direction. SolutionFinding the Dot Product of Two Vectors
As we discussed earlier in the section, scalar multiplication involves multiplying a vector by a scalar, and the result is a vector. As we have seen, multiplying a vector by a number is called scalar multiplication. If we multiply a vector by a vector, there are two possibilities: the dot product and the cross product. We will only examine the dot product here; you may encounter the cross product in more advanced mathematics courses. The dot product of two vectors involves multiplying two vectors together, and the result is a scalar.A General Note: Dot Product
The dot product of two vectors and is the sum of the product of the horizontal components and the product of the vertical components.
To find the angle between the two vectors, use the formula below.
Example 14: Finding the Dot Product of Two Vectors
Find the dot product of and .Solution
Using the formula, we haveExample 15: Finding the Dot Product of Two Vectors and the Angle between Them
Find the dot product of v1 = 5i + 2j and v2 = 3i + 7j. Then, find the angle between the two vectors.Solution
Finding the dot product, we multiply corresponding components.
To find the angle between them, we use the formula .

Example 16: Finding the Angle between Two Vectors
Find the angle between and .Solution
Using the formula, we have
Example 17: Finding Ground Speed and Bearing Using Vectors
We now have the tools to solve the problem we introduced in the opening of the section. An airplane is flying at an airspeed of 200 miles per hour headed on a SE bearing of 140°. A north wind (from north to south) is blowing at 16.2 miles per hour. What are the ground speed and actual bearing of the plane?
Solution
The ground speed is represented by in the diagram, and we need to find the angle in order to calculate the adjusted bearing, which will be . Notice in Figure 19, that angle must be equal to angle by the rule of alternating interior angles, so angle is 140°. We can find by the Law of Cosines:
The ground speed is approximately 213 miles per hour. Now we can calculate the bearing using the Law of Sines.
Therefore, the plane has a SE bearing of 140°+2.8°=142.8°. The ground speed is 212.7 miles per hour.
Key Concepts
- The position vector has its initial point at the origin.
- If the position vector is the same for two vectors, they are equal.
- Vectors are defined by their magnitude and direction.
- If two vectors have the same magnitude and direction, they are equal.
- Vector addition and subtraction result in a new vector found by adding or subtracting corresponding elements.
- Scalar multiplication is multiplying a vector by a constant. Only the magnitude changes; the direction stays the same.
- Vectors are comprised of two components: the horizontal component along the positive x-axis, and the vertical component along the positive y-axis.
- The unit vector in the same direction of any nonzero vector is found by dividing the vector by its magnitude.
- The magnitude of a vector in the rectangular coordinate system is .
- In the rectangular coordinate system, unit vectors may be represented in terms of and where represents the horizontal component and represents the vertical component. Then, v = ai + bj is a scalar multiple of by real numbers .
- Adding and subtracting vectors in terms of i and j consists of adding or subtracting corresponding coefficients of i and corresponding coefficients of j.
- A vector v = ai + bj is written in terms of magnitude and direction as .
- The dot product of two vectors is the product of the terms plus the product of the terms.
- We can use the dot product to find the angle between two vectors.
- Dot products are useful for many types of physics applications.
Glossary
- dot product
- given two vectors, the sum of the product of the horizontal components and the product of the vertical components
- initial point
- the origin of a vector
- magnitude
- the length of a vector; may represent a quantity such as speed, and is calculated using the Pythagorean Theorem
- resultant
- a vector that results from addition or subtraction of two vectors, or from scalar multiplication
- scalar
- a quantity associated with magnitude but not direction; a constant
- scalar multiplication
- the product of a constant and each component of a vector
- standard position
- the placement of a vector with the initial point at and the terminal point , represented by the change in the x-coordinates and the change in the y-coordinates of the original vector
- terminal point
- the end point of a vector, usually represented by an arrow indicating its direction
- unit vector
- a vector that begins at the origin and has magnitude of 1; the horizontal unit vector runs along the x-axis and is defined as the vertical unit vector runs along the y-axis and is defined as .
- vector
- a quantity associated with both magnitude and direction, represented as a directed line segment with a starting point (initial point) and an end point (terminal point)
- vector addition
- the sum of two vectors, found by adding corresponding components
Section Exercises
1. What are the characteristics of the letters that are commonly used to represent vectors? 2. How is a vector more specific than a line segment? 3. What are i and j, and what do they represent? 4. What is component form? 5. When a unit vector is expressed as , which letter is the coefficient of the i and which the j? 6. Given a vector with initial point (5,2) and terminal point (−1,−3), find an equivalent vector whose initial point is (0,0). Write the vector in component form . 7. Given a vector with initial point (−4,2) and terminal point (3,−3), find an equivalent vector whose initial point is (0,0). Write the vector in component form . 8. Given a vector with initial point (7,−1) and terminal point (−1,−7), find an equivalent vector whose initial point is (0,0). Write the vector in component form . For the following exercises, determine whether the two vectors u and v are equal, where u has an initial point and a terminal point and v has an initial point and a terminal point . 9. , and 10. , and 11. , and 12. , and 13. , and 14. Given initial point and terminal point , write the vector v in terms of i and j. 15. Given initial point and terminal point , write the vector v in terms of i and j. For the following exercises, use the vectors , , and . 16. Find 17. Find For the following exercises, use the given vectors to compute , , and . 18. 19. 20. Let . Find a vector that is half the length and points in the same direction as v. 21. Let . Find a vector that is twice the length and points in the opposite direction as v. For the following exercises, find a unit vector in the same direction as the given vector. 22. 23. 24. 25. 26. 27. For the following exercises, find the magnitude and direction of the vector, . 28. 29. 30. 31. 32. Given and , calculate . 33. Given and , calculate . 34. Given and , calculate . 35. Given and , calculate . For the following exercises, given v, draw v, 3v, and . 36. 37. 38. For the following exercises, use the vectors shown to sketch , , and . 39.








Find the magnitude of the normal (perpendicular) component of the force.
Find the magnitude of the component of the force that is parallel to the ramp.
56. A 25-pound box is resting on a ramp that is inclined 8°. Rounding to the nearest tenth,
Find the magnitude of the normal (perpendicular) component of the force.
Find the magnitude of the component of the force that is parallel to the ramp.
57. Find the magnitude of the horizontal and vertical components of a vector with magnitude 8 pounds pointed in a direction of above the horizontal. Round to the nearest hundredth.
58. Find the magnitude of the horizontal and vertical components of the vector with magnitude 4 pounds pointed in a direction of above the horizontal. Round to the nearest hundredth.
59. Find the magnitude of the horizontal and vertical components of the vector with magnitude 5 pounds pointed in a direction of above the horizontal. Round to the nearest hundredth.
60. Find the magnitude of the horizontal and vertical components of the vector with magnitude 1 pound pointed in a direction of above the horizontal. Round to the nearest hundredth.
61. A woman leaves home and walks 3 miles west, then 2 miles southwest. How far from home is she, and in what direction must she walk to head directly home?
62. A boat leaves the marina and sails 6 miles north, then 2 miles northeast. How far from the marina is the boat, and in what direction must it sail to head directly back to the marina?
63. A man starts walking from home and walks 4 miles east, 2 miles southeast, 5 miles south, 4 miles southwest, and 2 miles east. How far has he walked? If he walked straight home, how far would he have to walk?
64. A woman starts walking from home and walks 4 miles east, 7 miles southeast, 6 miles south, 5 miles southwest, and 3 miles east. How far has she walked? If she walked straight home, how far would she have to walk?
65. A man starts walking from home and walks 3 miles at north of west, then 5 miles at west of south, then 4 miles at north of east. If he walked straight home, how far would he have to the walk, and in what direction?
66. A woman starts walking from home and walks 6 miles at north of east, then 2 miles at east of south, then 5 miles at south of west. If she walked straight home, how far would she have to walk, and in what direction?
67. An airplane is heading north at an airspeed of 600 km/hr, but there is a wind blowing from the southwest at 80 km/hr. How many degrees off course will the plane end up flying, and what is the plane’s speed relative to the ground?
68. An airplane is heading north at an airspeed of 500 km/hr, but there is a wind blowing from the northwest at 50 km/hr. How many degrees off course will the plane end up flying, and what is the plane’s speed relative to the ground?
69. An airplane needs to head due north, but there is a wind blowing from the southwest at 60 km/hr. The plane flies with an airspeed of 550 km/hr. To end up flying due north, how many degrees west of north will the pilot need to fly the plane?
70. An airplane needs to head due north, but there is a wind blowing from the northwest at 80 km/hr. The plane flies with an airspeed of 500 km/hr. To end up flying due north, how many degrees west of north will the pilot need to fly the plane?
71. As part of a video game, the point (5,7) is rotated counterclockwise about the origin through an angle of . Find the new coordinates of this point.
72. As part of a video game, the point (7,3) is rotated counterclockwise about the origin through an angle of . Find the new coordinates of this point.
73. Two children are throwing a ball back and forth straight across the back seat of a car. The ball is being thrown 10 mph relative to the car, and the car is traveling 25 mph down the road. If one child doesn't catch the ball, and it flies out the window, in what direction does the ball fly (ignoring wind resistance)?
74. Two children are throwing a ball back and forth straight across the back seat of a car. The ball is being thrown 8 mph relative to the car, and the car is traveling 45 mph down the road. If one child doesn't catch the ball, and it flies out the window, in what direction does the ball fly (ignoring wind resistance)?
75. A 50-pound object rests on a ramp that is inclined 19°. Find the magnitude of the components of the force parallel to and perpendicular to (normal) the ramp to the nearest tenth of a pound.
76. Suppose a body has a force of 10 pounds acting on it to the right, 25 pounds acting on it upward, and 5 pounds acting on it 45° from the horizontal. What single force is the resultant force acting on the body?
77. Suppose a body has a force of 10 pounds acting on it to the right, 25 pounds acting on it ─135° from the horizontal, and 5 pounds acting on it directed 150° from the horizontal. What single force is the resultant force acting on the body?
78. The condition of equilibrium is when the sum of the forces acting on a body is the zero vector. Suppose a body has a force of 2 pounds acting on it to the right, 5 pounds acting on it upward, and 3 pounds acting on it 45° from the horizontal. What single force is needed to produce a state of equilibrium on the body?
79. Suppose a body has a force of 3 pounds acting on it to the left, 4 pounds acting on it upward, and 2 pounds acting on it from the horizontal. What single force is needed to produce a state of equilibrium on the body? Draw the vector.