Vectors in Space

8 Vectors in the Plane

Learning Objectives

• Describe a plane vector, using correct notation.
• Perform basic vector operations (scalar multiplication, addition, subtraction).
• Express a vector in component form.
• Explain the formula for the magnitude of a vector.
• Express a vector in terms of unit vectors.
• Give two examples of vector quantities.

When describing the movement of an airplane in flight, it is important to communicate two pieces of information: the direction in which the plane is traveling and the plane’s speed. When measuring a force, such as the thrust of the plane’s engines, it is important to describe not only the strength of that force, but also the direction in which it is applied. Some quantities, such as or force, are defined in terms of both size (also called magnitude) and direction. A quantity that has magnitude and direction is called a vector. In this text, we denote vectors by boldface letters, such as v.

Definition

A vector is a quantity that has both magnitude and direction.

Vector Representation

A vector in a plane is represented by a directed line segment (an arrow). The endpoints of the segment are called the initial point and the terminal point of the vector. An arrow from the initial point to the terminal point indicates the direction of the vector. The length of the line segment represents its magnitude. We use the notation to denote the magnitude of the vector A vector with an initial point and terminal point that are the same is called the zero vector, denoted The zero vector is the only vector without a direction, and by convention can be considered to have any direction convenient to the problem at hand.

Vectors with the same magnitude and direction are called equivalent vectors. We treat equivalent vectors as equal, even if they have different initial points. Thus, if and are equivalent, we write

Definition

Vectors are said to be equivalent vectors if they have the same magnitude and direction.

The arrows in (Figure)(b) are equivalent. Each arrow has the same length and direction. A closely related concept is the idea of parallel vectors. Two vectors are said to be parallel if they have the same or opposite directions. We explore this idea in more detail later in the chapter. A vector is defined by its magnitude and direction, regardless of where its initial point is located.

(a) A vector is represented by a directed line segment from its initial point to its terminal point. (b) Vectors through are equivalent.

The use of boldface, lowercase letters to name vectors is a common representation in print, but there are alternative notations. When writing the name of a vector by hand, for example, it is easier to sketch an arrow over the variable than to simulate boldface type: When a vector has initial point and terminal point the notation is useful because it indicates the direction and location of the vector.

Sketching Vectors

Sketch a vector in the plane from initial point to terminal point

See (Figure). Because the vector goes from point to point we name it

The vector with initial point and terminal point is named

Sketch the vector where is point and is point

Hint

The first point listed in the name of the vector is the initial point of the vector.

Combining Vectors

Vectors have many real-life applications, including situations involving force or velocity. For example, consider the forces acting on a boat crossing a river. The boat’s motor generates a force in one direction, and the current of the river generates a force in another direction. Both forces are vectors. We must take both the magnitude and direction of each force into account if we want to know where the boat will go.

A second example that involves vectors is a quarterback throwing a football. The quarterback does not throw the ball parallel to the ground; instead, he aims up into the air. The velocity of his throw can be represented by a vector. If we know how hard he throws the ball (magnitude—in this case, speed), and the angle (direction), we can tell how far the ball will travel down the field.

A real number is often called a scalar in mathematics and physics. Unlike vectors, scalars are generally considered to have a magnitude only, but no direction. Multiplying a vector by a scalar changes the vector’s magnitude. This is called scalar multiplication. Note that changing the magnitude of a vector does not indicate a change in its direction. For example, wind blowing from north to south might increase or decrease in speed while maintaining its direction from north to south.

Definition

The product of a vector v and a scalar k is a vector with a magnitude that is times the magnitude of and with a direction that is the same as the direction of if and opposite the direction of if This is called scalar multiplication. If or then

As you might expect, if we denote the product as

Note that has the same magnitude as but has the opposite direction ((Figure)).

(a) The original vector v has length n units. (b) The length of equals units. (c) The length of is units. (d) The vectors and have the same length but opposite directions.

Another operation we can perform on vectors is to add them together in vector addition, but because each vector may have its own direction, the process is different from adding two numbers. The most common graphical method for adding two vectors is to place the initial point of the second vector at the terminal point of the first, as in (Figure)(a). To see why this makes sense, suppose, for example, that both vectors represent displacement. If an object moves first from the initial point to the terminal point of vector then from the initial point to the terminal point of vector the overall displacement is the same as if the object had made just one movement from the initial point to the terminal point of the vector For obvious reasons, this approach is called the triangle method. Notice that if we had switched the order, so that was our first vector and v was our second vector, we would have ended up in the same place. (Again, see (Figure)(a).) Thus,

A second method for adding vectors is called the parallelogram method. With this method, we place the two vectors so they have the same initial point, and then we draw a parallelogram with the vectors as two adjacent sides, as in (Figure)(b). The length of the diagonal of the parallelogram is the sum. Comparing (Figure)(b) and (Figure)(a), we can see that we get the same answer using either method. The vector is called the vector sum.

Definition

The sum of two vectors and can be constructed graphically by placing the initial point of at the terminal point of Then, the vector sum, is the vector with an initial point that coincides with the initial point of and has a terminal point that coincides with the terminal point of This operation is known as vector addition.

(a) When adding vectors by the triangle method, the initial point of is the terminal point of (b) When adding vectors by the parallelogram method, the vectors and have the same initial point.

It is also appropriate here to discuss vector subtraction. We define as The vector is called the vector difference. Graphically, the vector is depicted by drawing a vector from the terminal point of to the terminal point of ((Figure)).

(a) The vector difference is depicted by drawing a vector from the terminal point of to the terminal point of (b) The vector is equivalent to the vector

In (Figure)(a), the initial point of is the initial point of The terminal point of is the terminal point of These three vectors form the sides of a triangle. It follows that the length of any one side is less than the sum of the lengths of the remaining sides. So we have

This is known more generally as the triangle inequality. There is one case, however, when the resultant vector has the same magnitude as the sum of the magnitudes of and This happens only when and have the same direction.

Combining Vectors

Given the vectors and shown in (Figure), sketch the vectors

1. Vectors and lie in the same plane.

1. The vector has the same direction as it is three times as long as

Vector has the same direction as and is three times as long.
2. Use either addition method to find
To find align the vectors at their initial points or place the initial point of one vector at the terminal point of the other. (a) The vector is the diagonal of the parallelogram with sides and (b) The vector is the third side of a triangle formed with placed at the terminal point of

3. To find we can first rewrite the expression as Then we can draw the vector then add it to the vector

Using vectors and from (Figure), sketch the vector

Hint

First sketch vectors and

Vector Components

Working with vectors in a plane is easier when we are working in a coordinate system. When the initial points and terminal points of vectors are given in Cartesian coordinates, computations become straightforward.

Comparing Vectors

Are and equivalent vectors?

1. has initial point and terminal point
has initial point and terminal point
2. has initial point and terminal point
has initial point and terminal point
1. The vectors are each units long, but they are oriented in different directions. So and are not equivalent ((Figure)).
These vectors are not equivalent.

2. Based on (Figure), and using a bit of geometry, it is clear these vectors have the same length and the same direction, so and are equivalent.
These vectors are equivalent.

Which of the following vectors are equivalent?

Vectors and are equivalent.

Hint

Equivalent vectors have both the same magnitude and the same direction.

We have seen how to plot a vector when we are given an initial point and a terminal point. However, because a vector can be placed anywhere in a plane, it may be easier to perform calculations with a vector when its initial point coincides with the origin. We call a vector with its initial point at the origin a standard-position vector. Because the initial point of any vector in standard position is known to be we can describe the vector by looking at the coordinates of its terminal point. Thus, if vector v has its initial point at the origin and its terminal point at we write the vector in component form as

When a vector is written in component form like this, the scalars x and y are called the components of

Definition

The vector with initial point and terminal point can be written in component form as

The scalars and are called the components of

Recall that vectors are named with lowercase letters in bold type or by drawing an arrow over their name. We have also learned that we can name a vector by its component form, with the coordinates of its terminal point in angle brackets. However, when writing the component form of a vector, it is important to distinguish between and The first ordered pair uses angle brackets to describe a vector, whereas the second uses parentheses to describe a point in a plane. The initial point of is the terminal point of is

When we have a vector not already in standard position, we can determine its component form in one of two ways. We can use a geometric approach, in which we sketch the vector in the coordinate plane, and then sketch an equivalent standard-position vector. Alternatively, we can find it algebraically, using the coordinates of the initial point and the terminal point. To find it algebraically, we subtract the x-coordinate of the initial point from the x-coordinate of the terminal point to get the x component, and we subtract the y-coordinate of the initial point from the y-coordinate of the terminal point to get the y component.

Rule: Component Form of a Vector

Let v be a vector with initial point and terminal point Then we can express v in component form as

Expressing Vectors in Component Form

Express vector with initial point and terminal point in component form.

1. Geometric
1. Sketch the vector in the coordinate plane ((Figure)).
2. The terminal point is 4 units to the right and 2 units down from the initial point.
3. Find the point that is 4 units to the right and 2 units down from the origin.
4. In standard position, this vector has initial point and terminal point
These vectors are equivalent.

2. Algebraic
In the first solution, we used a sketch of the vector to see that the terminal point lies 4 units to the right. We can accomplish this algebraically by finding the difference of the x-coordinates:

Similarly, the difference of the y-coordinates shows the vertical length of the vector.

So, in component form,

Vector has initial point and terminal point Express in component form.

Hint

You may use either geometric or algebraic method.

To find the magnitude of a vector, we calculate the distance between its initial point and its terminal point. The magnitude of vector is denoted or and can be computed using the formula

Note that because this vector is written in component form, it is equivalent to a vector in standard position, with its initial point at the origin and terminal point Thus, it suffices to calculate the magnitude of the vector in standard position. Using the distance formula to calculate the distance between initial point and terminal point we have

Based on this formula, it is clear that for any vector and if and only if

The magnitude of a vector can also be derived using the Pythagorean theorem, as in the following figure.

If you use the components of a vector to define a right triangle, the magnitude of the vector is the length of the triangle’s hypotenuse.

We have defined scalar multiplication and vector addition geometrically. Expressing vectors in component form allows us to perform these same operations algebraically.

Definition

Let and be vectors, and let be a scalar.

Scalar multiplication:

Performing Operations in Component Form

Let be the vector with initial point and terminal point and let

1. Express in component form and find Then, using algebra, find
2. and
1. To place the initial point of at the origin, we must translate the vector units to the left and units down ((Figure)). Using the algebraic method, we can express as

In component form,

2. To find add the x-components and the y-components separately:
3. To find multiply by the scalar
4. To find find and add it to

Let and let be the vector with initial point and terminal point

1. Find
2. Express in component form.
3. Find

a. b. c.

Hint

Use the Pythagorean Theorem to find To find start by finding the scalar multiples and

Now that we have established the basic rules of vector arithmetic, we can state the properties of vector operations. We will prove two of these properties. The others can be proved in a similar manner.

Properties of Vector Operations

Let be vectors in a plane. Let be scalars.

Proof of Commutative Property

Let and Apply the commutative property for real numbers:

Proof of Distributive Property

Apply the distributive property for real numbers:

Hint

Use the component form of the vectors.

We have found the components of a vector given its initial and terminal points. In some cases, we may only have the magnitude and direction of a vector, not the points. For these vectors, we can identify the horizontal and vertical components using trigonometry ((Figure)).

The components of a vector form the legs of a right triangle, with the vector as the hypotenuse.

Consider the angle formed by the vector v and the positive x-axis. We can see from the triangle that the components of vector are Therefore, given an angle and the magnitude of a vector, we can use the cosine and sine of the angle to find the components of the vector.

Finding the Component Form of a Vector Using Trigonometry

Find the component form of a vector with magnitude 4 that forms an angle of with the x-axis.

Let and represent the components of the vector ((Figure)). Then and The component form of the vector is

Use trigonometric ratios, and to identify the components of the vector.

Find the component form of vector with magnitude that forms an angle of with the positive x-axis.

Hint

and

Unit Vectors

A unit vector is a vector with magnitude For any nonzero vector we can use scalar multiplication to find a unit vector that has the same direction as To do this, we multiply the vector by the reciprocal of its magnitude:

Recall that when we defined scalar multiplication, we noted that For it follows that We say that is the unit vector in the direction of ((Figure)). The process of using scalar multiplication to find a unit vector with a given direction is called normalization.

The vector and associated unit vector In this case,

Finding a Unit Vector

Let

1. Find a unit vector with the same direction as
2. Find a vector with the same direction as such that
1. First, find the magnitude of then divide the components of by the magnitude:

2. The vector is in the same direction as and Use scalar multiplication to increase the length of without changing direction:

Let Find a vector with magnitude in the opposite direction as

Hint

First, find a unit vector in the same direction as

We have seen how convenient it can be to write a vector in component form. Sometimes, though, it is more convenient to write a vector as a sum of a horizontal vector and a vertical vector. To make this easier, let’s look at standard unit vectors. The standard unit vectors are the vectors and ((Figure)).

The standard unit vectors and

By applying the properties of vectors, it is possible to express any vector in terms of and in what we call a linear combination:

Thus, is the sum of a horizontal vector with magnitude and a vertical vector with magnitude as in the following figure.

The vector is the sum of and

Using Standard Unit Vectors
1. Express the vector in terms of standard unit vectors.
2. Vector is a unit vector that forms an angle of with the positive x-axis. Use standard unit vectors to describe
1. Resolve vector into a vector with a zero y-component and a vector with a zero x-component:
2. Because is a unit vector, the terminal point lies on the unit circle when the vector is placed in standard position ((Figure)).

The terminal point of lies on the unit circle

Let and let be a unit vector that forms an angle of with the positive x-axis. Express and in terms of the standard unit vectors.

Hint

Use sine and cosine to find the components of

Applications of Vectors

Because vectors have both direction and magnitude, they are valuable tools for solving problems involving such applications as motion and force. Recall the boat example and the quarterback example we described earlier. Here we look at two other examples in detail.

Finding Resultant Force

Jane’s car is stuck in the mud. Lisa and Jed come along in a truck to help pull her out. They attach one end of a tow strap to the front of the car and the other end to the truck’s trailer hitch, and the truck starts to pull. Meanwhile, Jane and Jed get behind the car and push. The truck generates a horizontal force of lb on the car. Jane and Jed are pushing at a slight upward angle and generate a force of lb on the car. These forces can be represented by vectors, as shown in (Figure). The angle between these vectors is Find the resultant force (the vector sum) and give its magnitude to the nearest tenth of a pound and its direction angle from the positive x-axis.

Two forces acting on a car in different directions.

To find the effect of combining the two forces, add their representative vectors. First, express each vector in component form or in terms of the standard unit vectors. For this purpose, it is easiest if we align one of the vectors with the positive x-axis. The horizontal vector, then, has initial point and terminal point It can be expressed as or

The second vector has magnitude and makes an angle of with the first, so we can express it as or Then, the sum of the vectors, or resultant vector, is and we have

The angle made by and the positive x-axis has so which means the resultant force has an angle of above the horizontal axis.

Finding Resultant Velocity

An airplane flies due west at an airspeed of mph. The wind is blowing from the northeast at mph. What is the ground speed of the airplane? What is the bearing of the airplane?

Let’s start by sketching the situation described ((Figure)).

Initially, the plane travels due west. The wind is from the northeast, so it is blowing to the southwest. The angle between the plane’s course and the wind is (Figure not drawn to scale.)

Set up a sketch so that the initial points of the vectors lie at the origin. Then, the plane’s velocity vector is The vector describing the wind makes an angle of with the positive x-axis:

When the airspeed and the wind act together on the plane, we can add their vectors to find the resultant force:

The magnitude of the resultant vector shows the effect of the wind on the ground speed of the airplane:

As a result of the wind, the plane is traveling at approximately mph relative to the ground.

To determine the bearing of the airplane, we want to find the direction of the vector

The overall direction of the plane is south of west.

An airplane flies due north at an airspeed of mph. The wind is blowing from the northwest at mph. What is the ground speed of the airplane?

Approximately mph

Hint

Sketch the vectors with the same initial point and find their sum.

Key Concepts

• Vectors are used to represent quantities that have both magnitude and direction.
• We can add vectors by using the parallelogram method or the triangle method to find the sum. We can multiply a vector by a scalar to change its length or give it the opposite direction.
• Subtraction of vectors is defined in terms of adding the negative of the vector.
• A vector is written in component form as
• The magnitude of a vector is a scalar:
• A unit vector has magnitude and can be found by dividing a vector by its magnitude: The standard unit vectors are A vector can be expressed in terms of the standard unit vectors as
• Vectors are often used in physics and engineering to represent forces and velocities, among other quantities.

For the following exercises, consider points and Determine the requested vectors and express each of them a. in component form and b. by using the standard unit vectors.

a. b.

a. b.

a. b.

a. b.

The unit vector in the direction of

a. b.

The unit vector in the direction of

A vector has initial point and terminal point Find the unit vector in the direction of Express the answer in component form.

A vector has initial point and terminal point Find the unit vector in the direction of Express the answer in component form.

The vector has initial point and terminal point that is on the y-axis and above the initial point. Find the coordinates of terminal point such that the magnitude of the vector is

The vector has initial point and terminal point that is on the x-axis and left of the initial point. Find the coordinates of terminal point such that the magnitude of the vector is

For the following exercises, use the given vectors and

1. Determine the vector sum and express it in both the component form and by using the standard unit vectors.
2. Find the vector difference and express it in both the component form and by using the standard unit vectors.
3. Verify that the vectors and and, respectively, and satisfy the triangle inequality.
4. Determine the vectors and Express the vectors in both the component form and by using standard unit vectors.

a. b. c. Answers will vary; d.

Let be a standard-position vector with terminal point Let be a vector with initial point and terminal point Find the magnitude of vector

Let be a standard-position vector with terminal point at Let be a vector with initial point and terminal point Find the magnitude of vector

Let and be two nonzero vectors that are nonequivalent. Consider the vectors and defined in terms of and Find the scalar such that vectors and are equivalent.

Let and be two nonzero vectors that are nonequivalent. Consider the vectors and defined in terms of and Find the scalars and such that vectors and are equivalent.

Consider the vector with components that depend on a real number As the number varies, the components of change as well, depending on the functions that define them.

1. Write the vectors and in component form.
2. Show that the magnitude of vector remains constant for any real number
3. As varies, show that the terminal point of vector describes a circle centered at the origin of radius

Consider vector with components that depend on a real number As the number varies, the components of change as well, depending on the functions that define them.

1. Write the vectors and in component form.
2. Show that the magnitude of vector remains constant for any real number
3. As varies, show that the terminal point of vector describes a circle centered at the origin of radius

Show that vectors and are equivalent for and where is an integer.

Show that vectors and are opposite for and where is an integer.

For the following exercises, find vector with the given magnitude and in the same direction as vector

For the following exercises, find the component form of vector given its magnitude and the angle the vector makes with the positive x-axis. Give exact answers when possible.

For the following exercises, vector is given. Find the angle that vector makes with the positive direction of the x-axis, in a counter-clockwise direction.

Let and be three nonzero vectors. If then show there are two scalars, and such that

Consider vectors and c = 0 Determine the scalars and such that

Let be a fixed point on the graph of the differential function with a domain that is the set of real numbers.

1. Determine the real number such that point is situated on the line tangent to the graph of at point
2. Determine the unit vector with initial point and terminal point

a. b.

Consider the function where

1. Determine the real number such that point s situated on the line tangent to the graph of at point
2. Determine the unit vector with initial point and terminal point

Consider and two functions defined on the same set of real numbers Let and be two vectors that describe the graphs of the functions, where Show that if the graphs of the functions and do not intersect, then the vectors and are not equivalent.

Find such that vectors and are equivalent.

Calculate the coordinates of point such that is a parallelogram, with and

Consider the points and Determine the component form of vector

The speed of an object is the magnitude of its related velocity vector. A football thrown by a quarterback has an initial speed of mph and an angle of elevation of Determine the velocity vector in mph and express it in component form. (Round to two decimal places.)

A baseball player throws a baseball at an angle of with the horizontal. If the initial speed of the ball is mph, find the horizontal and vertical components of the initial velocity vector of the baseball. (Round to two decimal places.)

A bullet is fired with an initial velocity of ft/sec at an angle of with the horizontal. Find the horizontal and vertical components of the velocity vector of the bullet. (Round to two decimal places.)

The horizontal and vertical components are ft/sec and ft/sec, respectively.

[T] A 65-kg sprinter exerts a force of N at a angle with respect to the ground on the starting block at the instant a race begins. Find the horizontal component of the force. (Round to two decimal places.)

[T] Two forces, a horizontal force of lb and another of lb, act on the same object. The angle between these forces is Find the magnitude and direction angle from the positive x-axis of the resultant force that acts on the object. (Round to two decimal places.)

The magnitude of resultant force is lb; the direction angle is

[T] Two forces, a vertical force of lb and another of lb, act on the same object. The angle between these forces is Find the magnitude and direction angle from the positive x-axis of the resultant force that acts on the object. (Round to two decimal places.)

[T] Three forces act on object. Two of the forces have the magnitudes N and N, and make angles and respectively, with the positive x-axis. Find the magnitude and the direction angle from the positive x-axis of the third force such that the resultant force acting on the object is zero. (Round to two decimal places.)

The magnitude of the third vector is N; the direction angle is

Three forces with magnitudes lb, lb, and lb act on an object at angles of and respectively, with the positive x-axis. Find the magnitude and direction angle from the positive x-axis of the resultant force. (Round to two decimal places.)

[T] An airplane is flying in the direction of east of north (also abbreviated as at a speed of mph. A wind with speed mph comes from the southwest at a bearing of What are the ground speed and new direction of the airplane?

The new ground speed of the airplane is mph; the new direction is

[T] A boat is traveling in the water at mph in a direction of (that is, east of north). A strong current is moving at mph in a direction of What are the new speed and direction of the boat?

[T] A 50-lb weight is hung by a cable so that the two portions of the cable make angles of and respectively, with the horizontal. Find the magnitudes of the forces of tension and in the cables if the resultant force acting on the object is zero. (Round to two decimal places.)

[T] A 62-lb weight hangs from a rope that makes the angles of and respectively, with the horizontal. Find the magnitudes of the forces of tension and in the cables if the resultant force acting on the object is zero. (Round to two decimal places.)

[T] A 1500-lb boat is parked on a ramp that makes an angle of with the horizontal. The boat’s weight vector points downward and is a sum of two vectors: a horizontal vector that is parallel to the ramp and a vertical vector that is perpendicular to the inclined surface. The magnitudes of vectors and are the horizontal and vertical component, respectively, of the boat’s weight vector. Find the magnitudes of and (Round to the nearest integer.)

lb, lb

[T] An 85-lb box is at rest on a incline. Determine the magnitude of the force parallel to the incline necessary to keep the box from sliding. (Round to the nearest integer.)

A guy-wire supports a pole that is ft high. One end of the wire is attached to the top of the pole and the other end is anchored to the ground ft from the base of the pole. Determine the horizontal and vertical components of the force of tension in the wire if its magnitude is lb. (Round to the nearest integer.)

The two horizontal and vertical components of the force of tension are lb and lb, respectively.

A telephone pole guy-wire has an angle of elevation of with respect to the ground. The force of tension in the guy-wire is lb. Find the horizontal and vertical components of the force of tension. (Round to the nearest integer.)

Glossary

component
a scalar that describes either the vertical or horizontal direction of a vector
equivalent vectors
vectors that have the same magnitude and the same direction
initial point
the starting point of a vector
magnitude
the length of a vector
normalization
using scalar multiplication to find a unit vector with a given direction
parallelogram method
a method for finding the sum of two vectors; position the vectors so they share the same initial point; the vectors then form two adjacent sides of a parallelogram; the sum of the vectors is the diagonal of that parallelogram
scalar
a real number
scalar multiplication
a vector operation that defines the product of a scalar and a vector
standard-position vector
a vector with initial point
standard unit vectors
unit vectors along the coordinate axes:
terminal point
the endpoint of a vector
triangle inequality
the length of any side of a triangle is less than the sum of the lengths of the other two sides
triangle method
a method for finding the sum of two vectors; position the vectors so the terminal point of one vector is the initial point of the other; these vectors then form two sides of a triangle; the sum of the vectors is the vector that forms the third side; the initial point of the sum is the initial point of the first vector; the terminal point of the sum is the terminal point of the second vector
unit vector
a vector with margnitude
vector
a mathematical object that has both magnitude and direction