Solutions to Scalar and Vector Problems

Example 1

A hiker walks 53.1 degrees north of east for 2.5km then due east for 2.0km. What is her total displacement from her starting point if you measure the distance along a straight line?

The displacement is measured in distance and angle so we will compute both.

First, we draw a diagram of the hiker's path.

Let A be the first leg of the journey and B be the second. What we are looking for is the vector A+B. We find that:

So the distance from the starting point is given by the pythagorean theorem: (For convenience let C be the resultant vector.)

The angle of the resultant vector is:

Example 2

A telephone pole support cable is in the way of some construction workers. In order for the work to proceed, the cable must be moved 2 meters closer to the pole. If the pole is 10 meters tall and the cable is currently fastened to the ground 8 meters from the pole, how much will the workers need to cut off from the cable when they move it?

Again, we draw a diagram to help clarify the problem.

The original cable length is

or 12.8m.

By moving the cable 2 meters closer to the pole, we shorten the overall length of the cable to

or 11.7m. Therefore, 1.1m must be cut off the cable.

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