8. Vectors

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

||2u||

Given vectors u and v, find: v - 3u. 

u = 2i, v = i + j

In Exercises 53–56, let u = -2i + 3j, v = 6i - j, w = -3i. Find each specified vector or scalar. ||u + v||² - ||u - v||²

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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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.

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c + d

If P₁ = (-2, 3), P₂ = (-1, 5), and v is the vector from P₁ to P₂, Write v in terms of i and j.

In Exercises 1–4, u and v have the same direction. In each exercise: Find ||v||.

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

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

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

In Exercises 31–32, find the unit vector that has the same direction as the vector v.

v = -i + 2j

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

〈6, -3〉

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