Multiple-Concept Example 9 deals with the concepts that are important in this problem. A grasshopper makes four jumps. The displacement vectors are (1) 27.0 cm, due west; (2) 23.0 cm, 35.0° south of west; (3) 28.0 cm, 55.0° south of east; and (4) 35.0 cm, 63.0° north of east. Find the magnitude and direction of the resultant displacement. Express the direction with respect to due west.

Concept Example 9

A jogger runs 145 m in a direction 20.0° east of north (displacement vector A_(vector)) and then 105 m in a direction 35.0° south of east (displacement vector B_(vector)). Using components, determine the magnitude and direction of the resultant vector C_(vector) for these two displacements.

Reasoning Figure 1.22 shows the vectors A_(vector) and B_(vector), assuming that the y axis corresponds to the direction due north. The vectors are arranged in a tail-to-head fashion, with the resultant vector C_(vector) drawn from the tail of A_(vector) to the head of B_(vector). The components of the vectors are also shown in the fi gure. Since C_(vector) and its components form a right triangle (red in the drawing), we will use the Pythagorean theorem and trigonometry to express the magnitude and directional angle θ for C_(vector) in terms of its components. The components of C_(vector) will then be obtained from the components of A_(vector) and B_(vector) and the data given for these two vectors.

Knowns and Unknowns The data for this problem are listed in the table that follows:

Description Symbol Value Comment Magnitude of vector A 145 m Direction of vector 20.0 east of north See Figure 1.22. Magnitude of vector B 105 m Direction of vector B 35.0 south of east See Figure 1.22. Unknown Variable Magnitude of resultant vector C Direction of resultant vector ?