College Physics by Openstax Chapter 3 Problem 7: Verification of Vector Subtraction through Alternative Walking Paths

Solution:

Part A

Consider Figure 3-7A

First, we solve for the value of α using a simple geometry.

\alpha = 40^\circ+\left( 90^\circ-20^\circ \right) = 110^\circ

We can solve for R using the cosine law.

\begin{align*} R^2 & = A^2 +B^2-2AB \cos \alpha \\ R & = \sqrt{A^2 +B^2-2AB \cos \alpha} \\ R & = \sqrt{\left( 12.0\ \text{m} \right)^2+\left( 20.0\ \text{m} \right)^2-2\left( 12.0\ \text{m} \right)\left( 20.0\ \text{m} \right) \cos 110^\circ} \\ R & = 26.6115 \ \text{m} \\ R & = 26.6 \ \text{m} \ \qquad \ {\color{DarkOrange} \left( \text{Answer} \right)} \end{align*}

Before we can solve for θ, we need to solve for 𝛽 first using the sine law.

\begin{align*} \frac{\sin \beta}{B} & = \frac{\sin \alpha }{R} \\ \sin \beta & = \frac{B \ \sin \alpha}{R} \\ \beta & = \sin ^{-1} \left( \frac{B \ \sin \alpha}{R} \right) \\ \beta & = \sin ^{-1} \left( \frac{20.0 \ \text{m}\ \sin 110^{\circ}}{26.6115 \ \text{m}} \right) \\ \beta & = 44.9290^\circ \end{align*}

Now, we can solve for θ.

\begin{align*} \theta & = \left( 90^\circ + 20^\circ \right)-\beta \\ \theta & = 110^\circ - 44.9290^\circ \\ \theta & = 65.071 \\ \theta & = 65.1 ^\circ \end{align*}

Therefore, the compass reading is

65.1^\circ, \text{North of East} \ \qquad \ {\color{DarkOrange} \left( \text{Answer} \right)}

Part B

Refer to Figure 3-7B

First, we solve for the value of α using a simple geometry.

\alpha = 40^\circ+\left( 90^\circ-20^\circ \right) = 110^\circ

We can solve for R using the cosine law.

\begin{align*} R^2 & = A^2 +B^2-2AB \cos \alpha \\ R & = \sqrt{A^2 +B^2-2AB \cos \alpha} \\ R & = \sqrt{\left( 12.0\ \text{m} \right)^2+\left( 20.0\ \text{m} \right)^2-2\left( 12.0\ \text{m} \right)\left( 20.0\ \text{m} \right) \cos 110^\circ} \\ R & = 26.6115 \ \text{m} \\ R & = 26.6 \ \text{m} \ \qquad \ {\color{DarkOrange} \left( \text{Answer} \right)} \end{align*}

Before we can solve for θ, we need to solve for 𝛽 first using the sine law.

\begin{align*} \frac{\sin \beta}{A} & = \frac{\sin \alpha }{R} \\ \sin \beta & = \frac{A \ \sin \alpha}{R} \\ \beta & = \sin ^{-1} \left( \frac{A \ \sin \alpha}{R} \right) \\ \beta & = \sin ^{-1} \left( \frac{12.0 \ \text{m}\ \sin 110^{\circ}}{26.6115 \ \text{m}} \right) \\ \beta & = 25.0708^\circ \end{align*}

Now, we can solve for θ.

\begin{align*} \theta & = 40^\circ + \beta \\ \theta & = 40^\circ + 25.0708^\circ\\ \theta & = 65.0708^\circ\\ \theta & = 65.1 ^\circ \end{align*}

Therefore, the compass reading is

65.1^\circ, \text{South of West} \ \qquad \ {\color{DarkOrange} \left( \text{Answer} \right)}

This is consistent with Part A because (A-B) = -(B-A).