**Example 2**

Find the endpoint of the vector that starts at the origin, has a length of 16, and makes an angle of 143° with the positive *x*-axis (measured counterclockwise).

If our vector had a length of 1, then the endpoint would be on the unit circle. The *y*-coordinate would be sin(143°) ≈ 0.602 and the *x*-coordinate would be cos(143°) ≈ –0.799. Because of similar triangles, all we need to find our endpoint is to multiply each of these coordinates by 16. Because 16(–0.799) = –12.784 and 16(0.602) = 9.632, the endpoint of this vector is (–12.784,9.632). This is illustrated in Figure 16.8.

In general, the endpoint of a vector with angle *θ* and length *L* is (*L* · cos(*θ*),*L* · sin(*θ*)).

**Component Vectors**

The components of a vector can be very useful in solving problems. When an object is being pushed or pulled, only the component that points in the direction of movement actually contributes to the movement.

For example, suppose a heavy weight sits on the ground. A person ties a rope to the weight and pulls so that the rope makes an angle of 40° with the ground. Suppose that the person pulls with 150 pounds of force, enough force to lift a 150-pound object (see Figure 16.9).

The vector with magnitude 150 pounds and 40° can be broken into two component vectors. The one in the *x*-direction is 150 · cos(40°) ≈ 115 pounds. The one in the *y*-direction is 150 · sin(40°) ≈ 96 pounds. This means that only 115 pounds of force is pulling the weight toward the person. The rest of the force is acting to lift the weight off the ground. Whether or not this is enough to make the weight move depends on how heavy it is and how much friction the floor has.

If the person were to pull with the same 150 pounds of force, but at a 30° angle instead, then the *x*-component of the vector would be 150 · cos(30°) ≈ 130 pounds of force. This is illustrated in Figure 16.10. This explains why people bend down low to push cars and pull heavy objects.

Similarly, if an object is on a slope, only a component of its weight acts to pull it down the slope.

**Example**

Suppose a 200-pound weight is put on a 15° ramp, as illustrated in Figure 16.11. How much force is pulling the weight down the slope?

Gravity is pulling the weight straight down. The direction down the slope is 15° from horizontal, thus 75° from straight down, as illustrated in Figure 16.12.

We want to know the length *x* of the component vector that points down the slope. This is adjacent to the 75° angle of a right triangle with hypotenuse 200. Thus,

Thus, about 51.8 pounds of the object's weight are pushing in the direction of the slope. If a person pushed up the slope with a force of more than 51.8 pounds, then the weight would slide uphill (after friction was overcome).

**Adding Vectors**

To add two vectors, convert them into components and add each component separately. For example, the sum of (8,3) and (–4,4) is (8 + (–4),3 + 4) = (4,7). If the two vectors represent two trips, the sum will be the result of traveling along first one path and then the second. This is illustrated in Figure 16.15.

If the two vectors represent forces, the sum represents the overall effect of applying both forces at the same time to a single object. This is illustrated in Figure 16.16.

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