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Vectors in Trigonometry Study Guide (page 3)

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Updated on Oct 2, 2011

Example 1

Suppose a hiker travels 15° west of north for 5 miles, and then 10° north of west for 8 miles. How far and in what direction is the hiker from the starting point?

The first vector has magnitude 5 and direction 105°, so the components are (5 · cos(105°),5 · sin(105°)) (–1.3,4.8). The second vector has magnitude 8 and direction 170°, so its components are (8 · cos(170°), 8 · sin(170°)) = (–7.9,1.4). The sum is thus (–1.3 - 7.9,4.8 + 1.4) = (–9.2,6.2). The magnitude of this vector is √(–9.2)2 + (6.2)2 = √123.08 ≈ 11.1, so the hiker is now 11.1 miles away from the starting point. The angle θ this vector makes with the x-axis is . Looking at the sketch in Figure 16.17, this direction is 34° north of west.

Figure 16.17

Example 2

Two people pull on a sack of potatoes. The angle between them is 40°. One person pulls with 100 pounds of force and the other with 70 pounds. In which direction will the sack move and with how much force?

To make things easier, we can suppose that the larger force is headed in the 0° direction. We thus have the situation in Figure 16.18.

Figure 16.18

The component form for the larger vector is (100,0). The component form of the second vector is (70 · cos(40°),70 · sin(40°)) = (53.6,45). The sum is thus (153.6,45). This has a magnitude of √(153.6)2 + 452 = √25,617.96 ≈ 160 pounds. The angle of this vector is .

Thus, the sack will move in a direction between the two people that is 16.3° from the person who is pulling with 100 pounds of force. It will move as if pulled in that direction by 160 pounds of weight.

Practice problems for this study guide can be found at:

Vectors in Trigonometry Practice Questions

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