# 2.4 The cross product  (Page 8/16)

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$\text{u}=⟨3,-1,2⟩,$ $\text{v}=⟨-2,0,1⟩$

$\text{w}=-\frac{1}{3\sqrt{6}}\text{i}-\frac{7}{3\sqrt{6}}\text{j}-\frac{2}{3\sqrt{6}}\text{k}$

$\text{u}=⟨2,6,1⟩,$ $\text{v}=⟨3,0,1⟩$

$\text{u}=\stackrel{\to }{AB},$ $\text{v}=\stackrel{\to }{AC},$ where $A\left(1,0,1\right),$ $B\left(1,-1,3\right),$ and $C\left(0,0,5\right)$

$\text{w}=-\frac{4}{\sqrt{21}}\text{i}-\frac{2}{\sqrt{21}}\text{j}-\frac{1}{\sqrt{21}}\text{k}$

$\text{u}=\stackrel{\to }{OP},$ $\text{v}=\stackrel{\to }{PQ},$ where $P\left(-1,1,0\right)$ and $Q\left(0,2,1\right)$

Determine the real number $\alpha$ such that $\text{u}\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}\text{v}$ and $\text{i}$ are orthogonal, where $\text{u}=3\text{i}+\text{j}-5\text{k}$ and $\text{v}=4\text{i}-2\text{j}+\alpha \text{k}.$

$\alpha =10$

Show that $\text{u}\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}\text{v}$ and $2\text{i}-14\text{j}+2\text{k}$ cannot be orthogonal for any $\alpha$ real number, where $\text{u}=\text{i}+7\text{j}-\text{k}$ and $\text{v}=\alpha \text{i}+5\text{j}+\text{k}.$

Show that $\text{u}\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}\text{v}$ is orthogonal to $\text{u}+\text{v}$ and $\text{u}-\text{v},$ where $\text{u}$ and $\text{v}$ are nonzero vectors.

Show that $\text{v}\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}\text{u}$ is orthogonal to $\left(\text{u}·\text{v}\right)\left(\text{u}+\text{v}\right)+\mathbf{\text{u}},$ where $\text{u}$ and $\text{v}$ are nonzero vectors.

Calculate the determinant $|\begin{array}{ccc}\text{i}\hfill & \hfill \text{j}& \hfill \text{k}\\ 1\hfill & \hfill -1& \hfill 7\\ 2\hfill & \hfill 0& \hfill 3\end{array}|.$

$-3\text{i}+11\text{j}+2\text{k}$

Calculate the determinant $|\begin{array}{ccc}\text{i}\hfill & \hfill \text{j}& \hfill \text{k}\\ 0\hfill & \hfill 3& \hfill -4\\ 1\hfill & \hfill 6& \hfill -1\end{array}|.$

For the following exercises, the vectors $\text{u}$ and $\text{v}$ are given. Use determinant notation to find vector $\text{w}$ orthogonal to vectors $\text{u}$ and $\text{v}.$

$\text{u}=⟨-1,0,{e}^{t}⟩,$ $\text{v}=⟨1,{e}^{\text{−}t},0⟩,$ where $t$ is a real number

$\text{w}=⟨-1,{e}^{t},\text{−}{e}^{\text{−}t}⟩$

$\text{u}=⟨1,0,x⟩,$ $\text{v}=⟨\frac{2}{x},1,0⟩,$ where $x$ is a nonzero real number

Find vector $\left(\text{a}-2\mathbf{\text{b}}\right)\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}\mathbf{\text{c}},$ where $\text{a}=|\begin{array}{ccc}\text{i}\hfill & \hfill \text{j}& \hfill \text{k}\\ 2\hfill & \hfill -1& \hfill 5\\ 0\hfill & \hfill 1& \hfill 8\end{array}|,$ $\mathbf{\text{b}}=|\begin{array}{ccc}\mathbf{\text{i}}\hfill & \hfill \mathbf{\text{j}}& \hfill \mathbf{\text{k}}\\ 0\hfill & \hfill 1& \hfill 1\\ 2\hfill & \hfill -1& \hfill -2\end{array}|,$ and $\mathbf{\text{c}}=\text{i}+\text{j}+\mathbf{\text{k}}.$

$-26\text{i}+17\text{j}+9\text{k}$

Find vector $\mathbf{\text{c}}\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}\left(\text{a}+3\mathbf{\text{b}}\right),$ where $\text{a}=|\begin{array}{ccc}\text{i}\hfill & \text{j}\hfill & \text{k}\hfill \\ 5\hfill & 0\hfill & 9\hfill \\ 0\hfill & 1\hfill & 0\hfill \end{array}|,$ $\mathbf{\text{b}}=|\begin{array}{ccc}\text{i}\hfill & \hfill \text{j}& \hfill \text{k}\\ 0\hfill & \hfill -1& \hfill 1\\ 7\hfill & \hfill 1& \hfill -1\end{array}|,$ and $\mathbf{\text{c}}=\text{i}-\text{k}.$

[T] Use the cross product $\text{u}\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}\text{v}$ to find the acute angle between vectors $\text{u}$ and $\text{v},$ where $\text{u}=\text{i}+2\text{j}$ and $\text{v}=\text{i}+\text{k}.$ Express the answer in degrees rounded to the nearest integer.

$72\text{°}$

[T] Use the cross product $\text{u}\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}\text{v}$ to find the obtuse angle between vectors $\text{u}$ and $\text{v},$ where $\text{u}=\text{−}\text{i}+3\text{j}+\text{k}$ and $\text{v}=\text{i}-2\text{j}.$ Express the answer in degrees rounded to the nearest integer.

Use the sine and cosine of the angle between two nonzero vectors $\text{u}$ and $\text{v}$ to prove Lagrange’s identity: ${‖\text{u}\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}\text{v}‖}^{2}={‖\text{u}‖}^{2}{‖\text{v}‖}^{2}-{\left(\text{u}·\text{v}\right)}^{2}.$

Verify Lagrange’s identity ${‖\text{u}\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}\text{v}‖}^{2}={‖\text{u}‖}^{2}{‖\text{v}‖}^{2}-{\left(\text{u}·\text{v}\right)}^{2}$ for vectors $\text{u}=\text{−}\text{i}+\text{j}-2\text{k}$ and $\text{v}=2\text{i}-\text{j}.$

Nonzero vectors $\text{u}$ and $\text{v}$ are called collinear if there exists a nonzero scalar $\alpha$ such that $\text{v}=\alpha \text{u}.$ Show that $\text{u}$ and $\text{v}$ are collinear if and only if $\text{u}\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}\text{v}=0.$

Nonzero vectors $\text{u}$ and $\text{v}$ are called collinear if there exists a nonzero scalar $\alpha$ such that $\text{v}=\alpha \text{u}.$ Show that vectors $\stackrel{\to }{AB}$ and $\stackrel{\to }{AC}$ are collinear, where $A\left(4,1,0\right),$ $B\left(6,5,-2\right),$ and $C\left(5,3,-1\right).$

Find the area of the parallelogram with adjacent sides $\text{u}=⟨3,2,0⟩$ and $\text{v}=⟨0,2,1⟩.$

$7$

Find the area of the parallelogram with adjacent sides $\text{u}=\text{i}+\text{j}$ and $\text{v}=\text{i}+\text{k}.$

Consider points $A\left(3,-1,2\right),B\left(2,1,5\right),$ and $C\left(1,-2,-2\right).$

1. Find the area of parallelogram $ABCD$ with adjacent sides $\stackrel{\to }{AB}$ and $\stackrel{\to }{AC}.$
2. Find the area of triangle $ABC.$
3. Find the distance from point $A$ to line $BC.$

a. $5\sqrt{6};$ b. $\frac{5\sqrt{6}}{2};$ c. $\frac{5\sqrt{6}}{\sqrt{59}}$

Consider points $A\left(2,-3,4\right),B\left(0,1,2\right),$ and $C\left(-1,2,0\right).$

1. Find the area of parallelogram $ABCD$ with adjacent sides $\stackrel{\to }{AB}$ and $\stackrel{\to }{AC}.$
2. Find the area of triangle $ABC.$
3. Find the distance from point $B$ to line $AC.$

In the following exercises, vectors $\text{u},\text{v},\text{and}\phantom{\rule{0.2em}{0ex}}\text{w}$ are given.

1. Find the triple scalar product $\text{u}·\left(\text{v}\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}\mathbf{\text{w}}\right).$
2. Find the volume of the parallelepiped with the adjacent edges $\text{u},\text{v},\text{and}\phantom{\rule{0.2em}{0ex}}\text{w}.$

$\text{u}=\text{i}+\text{j},$ $\text{v}=\text{j}+\text{k},$ and $\text{w}=\text{i}+\text{k}$

a. $2;$ b. $2$

$\text{u}=⟨-3,5,-1⟩,$ $\text{v}=⟨0,2,-2⟩,$ and $\text{w}=⟨3,1,1⟩$

Calculate the triple scalar products $\text{v}·\left(\text{u}\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}\mathbf{\text{w}}\right)$ and $\text{w}·\left(\text{u}\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}\text{v}\right),$ where $\text{u}=⟨1,1,1⟩,$ $\text{v}=⟨7,6,9⟩,$ and $\text{w}=⟨4,2,7⟩.$

$\text{v}·\left(\text{u}\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}\text{w}\right)=-1,$ $\text{w}·\left(\text{u}\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}\text{v}\right)=1$

Calculate the triple scalar products $\text{w}·\left(\text{v}\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}\text{u}\right)$ and $\text{u}·\left(\mathbf{\text{w}}\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}\text{v}\right),$ where $\text{u}=⟨4,2,-1⟩,$ $\text{v}=⟨2,5,-3⟩,$ and $\text{w}=⟨9,5,-10⟩.$

Find vectors $\text{a},\text{b},\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}\text{c}$ with a triple scalar product given by the determinant

$|\begin{array}{ccc}1\hfill & 2\hfill & 3\hfill \\ 0\hfill & 2\hfill & 5\hfill \\ 8\hfill & 9\hfill & 2\hfill \end{array}|.$ Determine their triple scalar product.

$\text{a}=⟨1,2,3⟩,$ $\mathbf{\text{b}}=⟨0,2,5⟩,$ $\mathbf{\text{c}}=⟨8,9,2⟩;$ $\text{a}·\left(\text{b}\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}\text{c}\right)=-9$

The triple scalar product of vectors $\text{a},\text{b},\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}\text{c}$ is given by the determinant

$|\begin{array}{ccc}\hfill 0& \hfill -2& \hfill 1\\ \hfill 0& \hfill 1& \hfill 4\\ \hfill 1& \hfill -3& \hfill 7\end{array}|.$ Find vector $\text{a}-\mathbf{\text{b}}+\mathbf{\text{c}}.$

Consider the parallelepiped with edges $OA,OB,$ and $OC,$ where $A\left(2,1,0\right),B\left(1,2,0\right),$ and $C\left(0,1,\alpha \right).$

1. Find the real number $\alpha >0$ such that the volume of the parallelepiped is $3$ units 3 .
2. For $\alpha =1,$ find the height $h$ from vertex $C$ of the parallelepiped. Sketch the parallelepiped.

a. $\alpha =1;$ b. $h=1,$

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