Log In Start studying!

Select your language

Suggested languages for you:

Americas

English (US)

Europe

Q. 13

Expert-verified

Calculus

Found in: Page 824

Book edition 1st

Author(s) Peter Kohn, Laura Taalman

Pages 1155 pages

ISBN 9781429241861

Answers without the blur.

Just sign up for free and you're in.

Register for Free I’ll do it later

Short Answer

Give an example of three nonzero vectors u, v and w in $ℝ^{3}$ such that $u \times v = u \times w$ but $v \neq w$ . What geometric relationship must the three vectors have for this to happen?

Let $u = (1, 0, 0), v = (2, 1, 1) and w = (4, 1, 1)$ .

If $u \times v = u \times w$ , then u is parallel to $v - w$ .

See the step by step solution

Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1. Given Information

Give an example of three nonzero vectors u, v and w in $ℝ^{3}$ such that role="math" localid="1649322061069" $u \times v = u \times w$ but $v \neq w$ . What geometric relationship must the three vectors have for this to happen?

Step 2. Let u=(1,0,0), v=(2,1,1) and w=(4,1,1)

Now finding the value of $u \times v$ .

$u \times v = \det [\begin{matrix} i & j & k \\ 1 & 0 & 0 \\ 2 & 1 & 1 \end{matrix}] u \times v = ((0) (1) - (1) (0)) i + ((1) (1) - (2) (0)) j + ((1) (1) - (2) (0)) k u \times v = (0 + 0) i + (1 - 0) j + (1 - 0) k u \times v = 0 i + 1 j + 1 k$

Step 3. Now finding the value of

$u \times w = \det [\begin{matrix} i & j & k \\ 1 & 0 & 0 \\ 4 & 1 & 1 \end{matrix}] u \times w = ((0) (1) - (1) (0)) i + ((1) (1) - (4) (0)) j + ((1) (1) - (4) (0)) k u \times w = (0 + 0) i + (1 - 0) j + (1 - 0) k u \times w = 0 i + 1 j + 1 k$

Hence, $u \times v = u \times w = 0 i + 1 j + 1 k$ but $v \neq w$ .

Step 4. Now finding the relation of three vectors.

If $u \times v = u \times w$ , then u is parallel to $v - w$ .

Most popular questions for Math Textbooks

Find $u + v a n d u - v$ also sketch $u, v, u + v, u - v$

role="math" localid="1649595165778" $u = (1, - 2) a n d v = (- 3, 6)$

Give an example of three vectors in $ℝ^{3}$ that form a right-handed triple. Explain how you can use the same three vectors to form a left-handed triple.

For each function f and value x = c in Exercises 35–44, use a sequence of approximations to estimate $f^{'} (c)$ . Illustrate your work with an appropriate sequence of graphs of secant lines.

role="math" localid="1648705352169" $f (x) = e^{x}, c = 0$

Find a vector of length 3 that points in the direction opposite to $<1, - 2, 3>$ .

Use limit rules and the continuity of power functions to prove that every polynomial function is continuous everywhere.

Want to see more solutions like these?

Sign up for free to discover our expert answers

Get Started - It’s free

Recommended explanations on Math Textbooks

94% of StudySmarter users get better grades.

Sign up for free