College Physics 3.16 – Resultant of two vectors

Solve the following problem using analytical techniques: Suppose you walk 18.0 m straight west and then 25.0 m straight north. How far are you from your starting point, and what is the compass direction of a line connecting your starting point to your final position? (If you represent the two legs of the walk as vector displacements A and B , as in Figure 3.60, then this problem asks you to find their sum R = A + B .)

The resultant R and θ are shown in the figure. We are given the vectors A and B, and the right triangle is formed.
Figure 3.60: The two displacements A and B add to give a total displacement R having magnitude R and direction θ.

Note that you can also solve this graphically. Discuss why the analytical technique for solving this problem is potentially more accurate than the graphical technique.


Consider the right triangle formed by the legs A, B, and R. We know that A is 18 m, B is 25 m, and we are solving for the magnitude of R. We can do this by using the Pythagorean Theorem. That is






So, the distance is about 30.8 meters from the starting point. To solve for the value of the unknown angle, θ, we can use the tangent function. That is 

tan\:\theta =\frac{B}{A}

tan\:\theta =\frac{25\:m}{18\:m}

\theta =tan^{-1}\left(\frac{25}{18}\right)

\theta =54.25^{\circ}

So, the compass reading, can be solved by taking the complimentary angle, \phi as shown in the figure.

The angle 𝜙 is the complimentary angle of 𝜃.
The angle ɸ is the complimentary angle of θ.

\phi =90^{\circ} -54.25^{\circ}

\phi =35.75^{\circ}

Therefore, the compass angle is 35.75^{\circ} \:West\:of\:North