8. Vectors

In Exercises 22–24, sketch each vector as a position vector and find its magnitude.

v = -3j

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.

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

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 = (4i - 2j) - (4i - 8j)

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

||w - u||

In Exercises 47–52, write the vector v in terms of i and j whose magnitude ||v|| and direction angle θ are given. ||v|| = 6, θ = 30°

For each pair of vectors u and v with angle θ between them, sketch the resultant.

|u| = 20, |v| = 30, θ = 30°

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

Solve each problem. See Examples 5 and 6.

Distance and Direction of a Motorboat A motorboat sets out in the direction N 80° 00′ E. The speed of the boat in still water is 20.0 mph. If the current is flowing directly south, and the actual direction of the motorboat is due east, find the speed of the current and the actual speed of the motorboat.

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

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

3w + 2v

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

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

v = 6i

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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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′