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Exercise 20

Avg difficulty 2.5

Let u⃗\vu and v⃗\vv be vectors in R5\R^5 with u⃗⋅v⃗=−1\vu \cdot \vv = -1, ∥u⃗∥=2\Vert \vu \Vert = 2, ∥v⃗∥=3\Vert \vv \Vert = 3. Use the properties of the dot product to find each of the following.

  1. u⃗⋅2v⃗\vu \cdot 2\vv

  2. v⃗⋅v⃗\vv \cdot \vv

  3. (u⃗+v⃗)⋅v⃗(\vu + \vv) \cdot \vv

  4. (2u⃗+4v⃗)⋅(u⃗−7v⃗)(2\vu+4\vv) \cdot (\vu - 7\vv)

  5. ∥u⃗∥∥v⃗∥cos⁡(θ)\Vert\vu\Vert \Vert\vv\Vert \cos(\theta), where θ\theta is the angle between u⃗\vu and v⃗\vv

  6. θ\theta

Long, Keller & Schlicker, Active Calculus Multivariable, §9.4 The Dot Product, Exercise 20

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Adapted from Active Calculus Multivariable © 2013–2026 Nicholas Long, Mitchel T. Keller, and Steven Schlicker. Licensed under CC BY-SA 4.0. Source: Active Calculus Multivariable. Imported at revision 692c03c.