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Q. 13

Expert-verified
Found in: Page 824

### Calculus

Book edition 1st
Author(s) Peter Kohn, Laura Taalman
Pages 1155 pages
ISBN 9781429241861

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

Let $u=\left(1,0,0\right),v=\left(2,1,1\right)\mathrm{and}w=\left(4,1,1\right)$.

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

See the step by step solution

## Step 1. Given Information

Give an example of three nonzero vectors u, v and w in ${\mathrm{ℝ}}^{3}$ such that role="math" localid="1649322061069" $u×v=u×w$ but $v\ne w$. 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 $u×v$.

$u×v=\mathrm{det}\left[\begin{array}{ccc}\mathrm{i}& \mathrm{j}& \mathrm{k}\\ 1& 0& 0\\ 2& 1& 1\end{array}\right]\phantom{\rule{0ex}{0ex}}u×v=\left(\left(0\right)\left(1\right)-\left(1\right)\left(0\right)\right)i+\left(\left(1\right)\left(1\right)-\left(2\right)\left(0\right)\right)j+\left(\left(1\right)\left(1\right)-\left(2\right)\left(0\right)\right)k\phantom{\rule{0ex}{0ex}}u×v=\left(0+0\right)i+\left(1-0\right)j+\left(1-0\right)k\phantom{\rule{0ex}{0ex}}u×v=0i+1j+1k$

## Step 3. Now finding the value of

$u×w=\mathrm{det}\left[\begin{array}{ccc}\mathrm{i}& \mathrm{j}& \mathrm{k}\\ 1& 0& 0\\ 4& 1& 1\end{array}\right]\phantom{\rule{0ex}{0ex}}u×w=\left(\left(0\right)\left(1\right)-\left(1\right)\left(0\right)\right)i+\left(\left(1\right)\left(1\right)-\left(4\right)\left(0\right)\right)j+\left(\left(1\right)\left(1\right)-\left(4\right)\left(0\right)\right)k\phantom{\rule{0ex}{0ex}}u×w=\left(0+0\right)i+\left(1-0\right)j+\left(1-0\right)k\phantom{\rule{0ex}{0ex}}u×w=0i+1j+1k$

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

## Step 4. Now finding the relation of three vectors.

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