Vectors in Trigonometry Study Guide
In this lesson, we explore the concept of a vector. We see how to break a vector down into component vectors—pieces that make for easy computation. We also explore a variety of word problems that are best solved by using vectors.
A vector is a length with a direction. Vectors are often depicted as arrows that point in the direction and have the correct length. The vector with a 2-inch length and northwest direction is shown in Figure 16.1.
In physics, vectors are used to represent forces. The arrow points in the direction of the push or pull, and the length represents the magnitude (strength) of the force. In navigation, vectors represent the components of journeys. The length represents the distance traveled in the direction of the arrow. Vectors can also represent the flow of wind and water, the stresses on objects, and anything else that has both a quantity and a direction.
There are several ways to represent a vector. One is to draw an arrow with the length and direction, as described above. Another is to give the length as a number and the direction as an angle. If we use the usual notation of 0° pointing straight to the right and positive angles measured counterclockwise, then the vector with length 40 and direction 200° would be as illustrated in Figure 16.2.
Usually, we consider that north is straight up (along the positive y-axis), south is straight down, east is to the right (along the positive x-axis), and west is to the left. Using these, the vector in Figure 16.2 is closest to pointing west. More specifically, it is 20° south of west. This is how the directions of navigation vectors are usually described (see Figure 16.3).
Unfortunately for us, navigators consider north to be 0° and then measure their angles clockwise, so northeast is 45°, east is 90°, south is 180°, and west is 270°. Because this conflicts with the way mathematicians measure angles (measured counterclockwise with east 0°), we shall stick with the "20° south of west" notation.
A last way to represent a vector is to put its start at the origin and give the coordinates of its endpoint. This is where trigonometry is used. The vector in Figure 16.3 can be used as the hypotenuse of a right triangle, as shown in Figure 16.4. The angle to the nearest axis, here 20°, is called the reference angle.
Thus, this vector can be described by the endpoint (–37.59,–13.68) or as a combination of the two component vectors shown in Figure 16.5. The vector can be broken into a component vector of 37.59 west, and another component of 13.68 south.
What is the length and direction of the vector that runs from the origin to the point (8,–15)?
We sketch the vector in Figure 16.6.
The length L of this vector can be found by the Pythagorean theorem.
L2 = 82 + 152, so L = √64 + 225 = √289 = 17
The reference angle θ is formed by the vector and the nearest axis, which in this case is the negative y-axis. The length opposite this angle is 8 and the adjacent length is 15, so
This vector has a length of 17 and points 28° east of south. We could also say that the angle of this vector is 298°. This is depicted in Figure 16.7.
In general, the length of the vector from the origin to (x,y) is L = √X2 + Y2. The reference angle can be found with an inverse tangent.
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