Chapter 10: Q. 75 (page 826)

Let u, v, and w be vectors in ${鈩�}^{3}$ with $u 鈮� 0$ . Show that if $u 脳 v = u 脳 w$ and $u 路 v = u 路 w$ , then $v = w$ .

Short Answer

Expert verified

If $u 脳 v = u 脳 w$ then $u 鈭� v - w$ .

If $u 路 v = u 路 w$ then u is orthogonal to $v 鈭� w$ this can only happen if $v - w = 0$

Step by step solution

Step 1. Given Information

Let u, v, and w be vectors in ${鈩�}^{3}$ with $u 鈮� 0$ . Show that if $u 脳 v = u 脳 w$ and $u 路 v = u 路 w$ , then $v = w$ .

Step 2. Let u, v, and w be vectors in ℝ3 with u≠0

$u 脳 v = u 脳 w Subtract u 脳 w on both side u 脳 v - u 脳 w = u 脳 w - u 脳 w u 脳 v - u 脳 w = 0 u 脳 (v - w) = 0 u = 0 v - w = 0$

That means $u 鈭� v - w$

Step 3. Now if u·v=u·w

If $u 路 v = u 路 w$ , then u is orthogonal to $v 鈭� w$ . Thus, $v 鈭� w$ is orthogonal to itself. This can only happen if $v 鈭� w = 0$ .

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91影视

Let u, v, and w be vectors in ${鈩�}^{3}$ with $u 鈮� 0$ . Show that if $u 脳 v = u 脳 w$ and $u 路 v = u 路 w$ , then $v = w$ .

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Let u, v, and w be vectors in ℝ3 with u≠0

Step 3. Now if u·v=u·w

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91影视

Let u, v, and w be vectors in 鈩�3with u鈮�0. Show that if u脳v=u脳wand u路v=u路w, then v=w.

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Let u, v, and w be vectors in ℝ3 with u≠0

Step 3. Now if u·v=u·w

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Applied Mathematics

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Study anywhere. Anytime. Across all devices.

Let u, v, and w be vectors in ${鈩�}^{3}$ with $u 鈮� 0$ . Show that if $u 脳 v = u 脳 w$ and $u 路 v = u 路 w$ , then $v = w$ .