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Answers without the blur. Sign up and see all textbooks for free! Q. 7

Expert-verified Found in: Page 823 ### Calculus

Book edition 1st
Author(s) Peter Kohn, Laura Taalman
Pages 1155 pages
ISBN 9781429241861 # If u and v are nonzero vectors in ${\mathrm{ℝ}}^{3}$, why do the equations role="math" localid="1649263352081" $u·\left(u×v\right)=0$ and $v·\left(u×v\right)=0$ tell us that the cross product is orthogonal to both u and v?

We prove that $u·\left(u×v\right)=0\mathrm{and}v·\left(u×v\right)=0$ tell us that the cross product is orthogonal to both u and v.

See the step by step solution

## Step 1. Given Information

If u and v are nonzero vectors in ${\mathrm{ℝ}}^{3}$, why do the equations $u·\left(u×v\right)=0$ and $v·\left(u×v\right)=0$ tell us that the cross product is orthogonal to both u and v?

## Step 2. u and v are nonzero vectors in ℝ3.

Let $u=\left({u}_{1},{u}_{2},{u}_{3}\right)\mathrm{and}v=\left({v}_{1},{v}_{2},{v}_{3}\right)$.

The determinant of a 3 × 3 matrix is

$u×v=\mathrm{det}\left[\begin{array}{ccc}\mathrm{i}& \mathrm{j}& \mathrm{k}\\ {\mathrm{u}}_{1}& {\mathrm{u}}_{2}& {\mathrm{u}}_{3}\\ {\mathrm{v}}_{1}& {\mathrm{v}}_{2}& {\mathrm{v}}_{3}\end{array}\right]$

## Step 3. Then, by the definition of the cross product,

$u×v=\left({u}_{2}{v}_{3}-{u}_{3}{v}_{2},{u}_{3}{v}_{1}-{u}_{1}{v}_{3},{u}_{1}{v}_{2}-{u}_{2}{v}_{1}\right)$

Now proving the equation $u·\left(u×v\right)=0$

$u·\left(u×v\right)=\left({u}_{1},{u}_{2},{u}_{3}\right)·\left({u}_{2}{v}_{3}-{u}_{3}{v}_{2},{u}_{3}{v}_{1}-{u}_{1}{v}_{3},{u}_{1}{v}_{2}-{u}_{2}{v}_{1}\right)\phantom{\rule{0ex}{0ex}}u·\left(u×v\right)={u}_{1}\left({u}_{2}{v}_{3}-{u}_{3}{v}_{2}\right)+{u}_{2}\left({u}_{3}{v}_{1}-{u}_{1}{v}_{3}\right)+{u}_{3}\left({u}_{1}{v}_{2}-{u}_{2}{v}_{1}\right)\phantom{\rule{0ex}{0ex}}u·\left(u×v\right)=0$

## Step 4. Now proving the equationv·(u×v)=0.

$v·\left(u×v\right)=\left({v}_{1},{v}_{2},{v}_{3}\right)·\left({u}_{2}{v}_{3}-{u}_{3}{v}_{2},{u}_{3}{v}_{1}-{u}_{1}{v}_{3},{u}_{1}{v}_{2}-{u}_{2}{v}_{1}\right)\phantom{\rule{0ex}{0ex}}v·\left(u×v\right)={v}_{1}\left({u}_{2}{v}_{3}-{u}_{3}{v}_{2}\right)+{v}_{2}\left({u}_{3}{v}_{1}-{u}_{1}{v}_{3}\right)+{v}_{3}\left({u}_{1}{v}_{2}-{u}_{2}{v}_{1}\right)\phantom{\rule{0ex}{0ex}}v·\left(u×v\right)=0$ ### Want to see more solutions like these? 