A spelunker is surveying a cave. She follows a passage 140 m straight west, then 210 m in a direction 45 east of south, and then 280 m at 30 east of north. After a fourth unmeasured displacement, she nds herself back where she started. Determine the magnitude and direction of the fourth displacement.

301.96 m with direction of 10.4 degrees north of west

Explanation:

Let i and j be the unit vector of east and north directions, respectively. So if she moving 140m straight west then her distance vector would be

When she walks 210 m in a direction 45 east of south, her displacement vector is:

When she walks 280 m in a direction 30 east of north, her displacement vector is:

Then she walk another passage to be back to the origin

where x4 and y4 are the displacement she made in the east and north direction in her 4th displacement:

So her last displacement vector is

This vector would have a displacement and direction of:

north of west