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Q. 13

Expert-verifiedFound in: Page 824

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 ${\mathrm{\mathbb{R}}}^{3}$ such that $u\times v=u\times w$ but $v\ne w$. What geometric relationship must the three vectors have for this to happen?

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

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

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

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

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

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

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

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

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