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Vectors

Jay Abramson

An airplane is flying at an airspeed of 200 miles per hour headed on a SE bearing of . A north wind (from north to south) is blowing at 16.2 miles per hour, as shown in Figure 1. What are the ground speed and actual bearing of the plane?

Figure 1

Ground speed refers to the speed of a plane relative to the ground. Airspeed refers to the speed a plane can travel relative to its surrounding air mass. These two quantities are not the same because of the effect of wind. In an earlier section, we used triangles to solve a similar problem involving the movement of boats. Later in this section, we will find the airplane’s groundspeed and bearing, while investigating another approach to problems of this type. First, however, let’s examine the basics of vectors.

1. 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 .

This last symbol has special significance. It is called the standard position. The position vector has an initial point and a terminal point . To change any vector into the position vector, we think about the change in the -coordinates and the change in the -coordinates. Thus, if the initial point of a vector is and the terminal point is , then the position vector is found by calculating

In Figure 2, we see the original vector and the position vector .

Figure 2

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 -coordinate from the other -coordinate, and one -coordinate from the other -coordinate. Thus,

The position vector begins at and terminates at . The graphs of both vectors are shown in Figure 3.

Figure 3

We see that the position vector is .

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 . See Figure 4.

Figure 4

2. 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.

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 -axis, or with the -axis, depending on the application. For a position vector, the direction is found by applying to the equation to get . The position vector, magnitude and direction are illustrated in Figure 5.

Figure 5

Two vectors and are considered equal if they have the same magnitude and the same direction. Additionally, if both vectors have the same position vector, they are equal.

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, as follows:

Next we find the magnitude .

Using the Pythagorean Theorem to find the magnitude to get:

The direction is found by applying to the equation . Thus we have

However, the angle terminates in the fourth quadrant, so we add to obtain a positive angle. Thus, . See Figure 6.

Figure 6

Example 4: Showing That Two Vectors Are Equal

Show that vector with initial point at and terminal point at is equal to vector with initial point at and terminal point at . Draw the position vector on the same grid as and . 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 and . We have and

Since the position vectors are the same, and are the same.

Figure 7

3. 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 and , or vector addition, produces a third vector , the resultant vector.

To find , we first draw the vector , and from the terminal end of , we drawn the vector . In other words, we have the initial point of meet the terminal end of . 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 is the resultant vector because it results from addition or subtraction of two vectors. The resultant vector travels directly from the beginning of to the end of in a straight path, as shown in Figure 8.

Figure 8

Vector subtraction is similar to vector addition. To find , view it as . Adding is reversing direction of and adding it to the end of . The new vector begins at the start of and stops at the end point of . See Figure 9 for a visual that compares vector addition and vector subtraction using parallelograms.

Figure 9

Example 5: Adding and Subtracting Vectors

Given and , find two new vectors , and .

Solution

To find the sum of two vectors, we add the components. Thus,

This is illustrated in Figure 10(a).

To find the difference of two vectors, add the negative components of to . Thus,

See Figure 10(b).