4.2 Vector addition and subtraction: graphical methods  (Page 3/15)

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 $\text{66.0º}$ north of east from her current location, and then travel 30.0 m in a direction $\text{112º}$ north of east (or $\text{22.0º}$ 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? Compare this location with the location of the dock.

Strategy

We can represent the first leg of the trip with a vector $\mathbf{\text{A}}$ , and the second leg of the trip with a vector $\mathbf{\text{B}}$ . The dock is located at a location $\mathbf{\text{A}}+\mathbf{\text{B}}$ . If the woman mistakenly travels in the opposite direction for the second leg of the journey, she will travel a distance $B$ (30.0 m) in the direction $\mathrm{180º}–\mathrm{112º}=\mathrm{68º}$ south of east. We represent this as $\mathbf{\text{–B}}$ , as shown below. The vector $\mathbf{\text{–B}}$ has the same magnitude as $\mathbf{\text{B}}$ but is in the opposite direction. Thus, she will end up at a location $\mathbf{\text{A}}+(\mathbf{\text{–B}})$ , or $\mathbf{\text{A}}–\mathbf{\text{B}}$ .

We will perform vector addition to compare the location of the dock, $\text{A }\text{+ }\mathbf{B}$ , with the location at which the woman mistakenly arrives, $\text{A + }(\text{–B})$ .

Solution

(1) To determine the location at which the woman arrives by accident, draw vectors $\mathbf{\text{A}}$ and $\mathbf{\text{–B}}$ .

(2) Place the vectors head to tail.

(3) Draw the resultant vector $\mathbf{R}$ .

(4) Use a ruler and protractor to measure the magnitude and direction of $\mathbf{R}$ .

In this case, $R=\text{23}\text{.}\text{0 m}$ and $\theta =7\text{.}\text{5º}$ south of east.

(5) To determine the location of the dock, we repeat this method to add vectors $\mathbf{\text{A}}$ and $\mathbf{\text{B}}$ . We obtain the resultant vector $\mathbf{\text{R}}\text{'}$ :

In this case $R\text{ = 52.9 m}$ and $\theta =\text{90.1º}$  north of east.

We can see that the woman will end up a significant distance from the dock if she travels in the opposite direction for the second leg of the trip.

Discussion

Because subtraction of a vector is the same as addition of a vector with the opposite direction, the graphical method of subtracting vectors works the same as for addition.