Repeat Exercise 3.16 using analytical techniques, but reverse the order of the two legs of the walk and show that you get the same final result. (This problem shows that adding them in reverse order gives the same result—that is, B + A = A + B .) Discuss how taking another path to reach the same point might help to overcome an obstacle blocking your other path.

Solution:

Considering the right triangle formed by the vectors A, B, and R. We can solve for the magnitude of R using the Pythagorean Theorem. That is

\begin{align*}
R & = \sqrt{A^2+B^2} \\
& = \sqrt{\left( 18.0 \text{m} \right)^2+\left( 25.0 \text{m} \right)^2} \\
& =30.806  \text{m} \\
& \approx 30.8  \text{m}  \qquad  {\color{DarkOrange} \left( \text{Answer} \right)}
\end{align*}

Then we solve for the compass direction by solving the value θ using the same right triangle.

\begin{align*}
\theta & = \arctan \left( \frac{B}{A} \right)  \\
& = \arctan \left( \frac{25.0 \text{m}}{18.0 \text{m}} \right) \\
& = 54.246 ^\circ \\
& \approx  54.2 ^ \circ \\
\end{align*}

Therefore, the compass direction of the resultant is 54.2° North of West.