McGraw Hill Glencoe Geometry, 2012
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McGraw Hill Glencoe Geometry, 2012 View details
7. Vectors
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Exercise 41 Page 607

Practice makes perfect
a We are given that Amy hiked due east for miles and then hiked south for miles. We're asked to draw a diagram describing the given situation, so let's start with drawing arrows describing Amy's journey.
As we can see, the component form of the vector representing Amy's position to the east is and the component form of the vector representing Amy's position to the south is Therefore, the resultant vector is the sum of these two vectors.
The vector represents the resultant position of Amy, which we will call
b In this part we are asked to determine how far and in what direction Amy is from her starting point. This distance is the magnitude of To evaluate this distance, we will use the Distance Formula. Let's substitute for the initial point and for the terminal point.
Simplify right-hand side
Amy's resultant distance is approximately miles. To evaluate the resultant direction, we need to find the measure of an angle that vector forms with a north-south line. We will call this angle

If we call the angle between the red and blue arrows then we can notice that and add to be

To find the measure of we can use one of the trigonometric ratios as the vectors form a right triangle. Let's recall that in a right triangle the tangent of an angle is the ratio between the leg opposite to this angle and the leg adjacent to this angle. Using this definition, we can write an equation.
Next, we can rewrite the equation using the inverse tangent to evaluate the measure of
Knowing the measure of we can find the measure of
The resultant direction is approximately east of south. Therefore, Amy is approximately miles at the angle of east of south from her starting point.