8. Vectors

Two tugboats are pulling a disabled speedboat into port with forces of 1240 lb and 1480 lb. The angle between these forces is 28.2°. Find the direction and magnitude of the equilibrant.

Two rescue vessels are pulling a broken-down motorboat toward a boathouse with forces of 840 lb and 960 lb. The angle between these forces is 24.5°. Find the direction and magnitude of the equilibrant.

Two forces of 692 newtons and 423 newtons act on a point. The resultant force is 786 newtons. Find the angle between the forces.

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

v - u

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.

<IMAGE>

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.

<IMAGE>

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?

<IMAGE>

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

〈6, -3〉

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

〈2, 0〉

Find the dot product for each pair of vectors.

〈6, -1〉, 〈2, 5〉