8. Vectors

Find the force required to keep a 75-lb sled from sliding down an incline that makes an angle of 27° with the horizontal. (Assume there is no friction.)

Use the figure to find each vector: u + v. Use vector notation as in Example 4.

CONCEPT PREVIEW Refer to vectors a through h below. Make a copy or a sketch of each vector, and then draw a sketch to represent each of the following. For example, find a + e by placing a and e so that their initial points coincide. Then use the parallelogram rule to find the resultant, as shown in the figure on the right.

2c

Let u = 〈-2, 1〉, v = 〈3, 4〉, and w = 〈-5, 12〉. Evaluate each expression.

(3u) • v

In Exercises 61–64, find the magnitude ||v||, to the nearest hundredth, and the direction angle θ, to the nearest tenth of a degree, for each given vector v. v = -10i + 15j

In Exercises 21–38, let u = 2i - 5j, v = -3i + 7j, and w = -i - 6j. Find each specified vector or scalar.

u - v

In Exercises 1–4, u and v have the same direction. In each exercise: Is u = v? Explain.

In Exercises 1–4, u and v have the same direction. In each exercise: Is u = v? Explain.

Two people are carrying a box. One person exerts a force of 150 lb at an angle of 62.4° with the horizontal. The other person exerts a force of 114 lb at an angle of 54.9°. Find the weight of the box.

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In Exercises 39–46, find the unit vector that has the same direction as the vector v.

v = 3i - 4j

Write each vector in the form 〈a, b〉. Write answers using exact values or to four decimal places, as appropriate.

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A luxury liner leaves port on a bearing of 110.0° and travels 8.8 mi. It then turns due west and travels 2.4 mi. How far is the liner from port, and what is its bearing from port?

Starting at point X, a ship sails 15.5 km on a bearing of 200°, then turns and sails 2.4 km on a bearing of 320°. Find the distance of the ship from point X.

Given vectors u and v, find: 2u + 3v. 

u = 〈-1, 2〉, v = 〈3, 0〉