b Amy is approximately 3.6 miles at the angle of 33.7^(∘) east of south from her starting point.
Practice makes perfect
a We are given that Amy hiked due east for 2 miles and then hiked south for 3 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 ⟨ 2,0⟩, and the component form of the vector representing Amy's position to the south is ⟨ 0, -3 ⟩. Therefore, the resultant vector is the sum of these two vectors.
⟨ 2,0 ⟩ + ⟨ 0, -3 ⟩ = ⟨ 2, -3⟩
The vector ⟨ 2,-3 ⟩ represents the resultant position of Amy, which we will call r.
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 r. To evaluate this distance, we will use the Distance Formula. Let's substitute ( , ) for the initial point and ( 2, -3) for the terminal point.
Amy's resultant distance is approximately 3.6 miles. To evaluate the resultant direction, we need to find the measure of an angle that vector r forms with a north-south line. We will call this angle α.
If we call the angle between the red and blue arrows θ, then we can notice that α and θ add to be 90^(∘).
To find the measure of θ, 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.
tan θ=3/2
Next, we can rewrite the equation using the inverse tangent to evaluate the measure of θ.
tan θ=3/2 ⇓
θ=tan ^(-1)3/2 ≈ 56.3^(∘)
Knowing the measure of θ, we can find the measure of α.
The resultant direction is approximately 33.7^(∘) east of south. Therefore, Amy is approximately 3.6 miles at the angle of 33.7^(∘) east of south from her starting point.
