Adding Vectors Graphically by Using the Head-to-Tail Method: A Woman Takes a Walk

Graphically represent each displacement vector with an arrow, labeling the first A, the second B, and the third C. Make the lengths proportional to the distance of the given displacement and orient the arrows as specified relative to an east-west line. Use the head-to-tail method outlined above to determine the magnitude and direction of the resultant displacement, which we’ll call R.

(1) Draw the three displacement vectors, creating a convenient scale (such as 1 cm of vector length on paper equals 1 m in the problem), as shown in Figure 5.8.

Figure 5.8 The three displacement vectors are drawn first.

(2) Place the vectors head to tail, making sure not to change their magnitude or direction, as shown in Figure 5.9.

Figure 5.9 Next, the vectors are placed head to tail.

(3) Draw the resultant vector R from the tail of the first vector to the head of the last vector, as shown in Figure 5.10.

Figure 5.10 The resultant vector is drawn .

(4) Use a ruler to measure the magnitude of R, remembering to convert back to the units of meters using the scale. Use a protractor to measure the direction of R. While the direction of the vector can be specified in many ways, the easiest way is to measure the angle between the vector and the nearest horizontal or vertical axis. Since R is south of the eastward pointing axis (the x-axis), we flip the protractor upside down and measure the angle between the eastward axis and the vector, as illustrated in Figure 5.11.

Figure 5.11 A ruler is used to measure the magnitude of R, and a protractor is used to measure the direction of R.

In this case, the total displacement R has a magnitude of 50 m and points $7^{\circ }$ south of east. Using its magnitude and direction, this vector can be expressed as

and