8. Vectors

Two forces of 128 lb and 253 lb act on a point. The resultant force is 320 lb. Find the angle between the forces.

A force of 176 lb makes an angle of 78° 50′ with a second force. The resultant of the two forces makes an angle of 41° 10′ with the first force. Find the magnitudes of the second force and of the resultant.

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Given u = 〈-2, 5〉 and v = 〈4, 3〉, find each of the following.

-5v

A force of 28.7 lb makes an angle of 42° 10′ with a second force. The resultant of the two forces makes an angle of 32° 40′ with the first force. Find the magnitudes of the second force and of the resultant.

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A force of 25 lb is required to hold an 80-lb crate on a hill. What angle does the hill make with the horizontal? 

Given u = 〈-2, 5〉 and v = 〈4, 3〉, find each of the following.

- 2u + 4v

Find the force required to keep a 3000-lb car parked on a hill that makes an angle of 15° with the horizontal.

To build the pyramids in Egypt, it is believed that giant causeways were constructed to transport the building materials to the site. One such causeway is said to have been 3000 ft long, with a slope of about 2.3°. How much force would be required to hold a 60-ton monolith on this causeway?

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Write each vector in the form a i + b j.

〈6, -3〉

Write each vector in the form a i + b j.

〈2, 0〉

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.

-b

Find the dot product for each pair of vectors.

4i, 5i - 9j

Find the magnitude and direction angle for each vector. Round angle measures to the nearest tenth, as necessary.

〈5, 7〉

Given vectors u and v, find: 2u + 3v.

u = 2i, v = i + j

Given vectors u and v, find: v - 3u. 

u = 2i, v = i + j