8. Vectors

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

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Use the parallelogram rule to find the magnitude of the resultant force for the two forces shown in each figure. Round answers to the nearest tenth.

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Two forces act on a point in the plane. The angle between the two forces is given. Find the magnitude of the resultant force.

forces of 250 and 450 newtons, forming an angle of 85°

Two forces act on a point in the plane. The angle between the two forces is given. Find the magnitude of the resultant force.

forces of 116 and 139 lb, forming an angle of 140° 50′

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.

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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.