Subtracting Vectors Graphically: A Woman Sailing a Boat

A woman sailing a boat at night is following directions to a dock. The instructions read to first sail 27.5 m in a direction ${66.0}^{\circ}$ north of east from her current location, and then travel 30.0 m in a direction $112^{\circ}$ north of east (or ${22.0}^{\circ}$ west of north). If the woman makes a mistake and travels in the opposite direction for the second leg of the trip, where will she end up? The two legs of the woman’s trip are illustrated in Figure 5.13.

An x-y axis is shown for reference. Vector A points upward from the x-axis is twenty-seven point five meters, and has an angle of sixty-six degrees. Vector B points upward from the x-axis is thirty meters, and has an angle of one hundred twelve degrees.

Figure 5.13 In the diagram, the first leg of the trip is represented by vector A and the second leg is represented by vector B.

Work it out on paper first. Then open the solution one step at a time, and stop as soon as you can finish on your own.

Strategy
We can represent the first leg of the trip with a vector A, and the second leg of the trip that she was supposed to take with a vector B. Since the woman mistakenly travels in the opposite direction for the second leg of the journey, the vector for second leg of the trip she actually takes is −B. Therefore, she will end up at a location A + (−B), or A − B. Note that −B has the same magnitude as B (30.0 m), but is in the opposite direction, $68^{\circ} (180^{\circ} - 112^{\circ})$ south of east, as illustrated in Figure 5.14.

Figure 5.14 Vector –B represents traveling in the opposite direction of vector B.

We use graphical vector addition to find where the woman arrives A + (−B).
Solution
(1) To determine the location at which the woman arrives by accident, draw vectors A and −B.

(2) Place the vectors head to tail.

(3) Draw the resultant vector R.

(4) Use a ruler and protractor to measure the magnitude and direction of R.

These steps are demonstrated in Figure 5.15.

Figure 5.15 The vectors are placed head to tail.

In this case

$R = 23.0 m$
5.3

and

$θ = 7.5 ° south of east.$
5.4

0 of 2 steps open

How did it go?

Strategy

Solution