# 2.4 The cross product  (Page 9/16)

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Consider points $A\left(\alpha ,0,0\right),B\left(0,\beta ,0\right),$ and $C\left(0,0,\gamma \right),$ with $\alpha ,$ $\beta ,$ and $\gamma$ positive real numbers.

1. Determine the volume of the parallelepiped with adjacent sides $\stackrel{\to }{OA},$ $\stackrel{\to }{OB},$ and $\stackrel{\to }{OC}.$
2. Find the volume of the tetrahedron with vertices $O,A,B,\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}C.$ ( Hint : The volume of the tetrahedron is $1\text{/}6$ of the volume of the parallelepiped.)
3. Find the distance from the origin to the plane determined by $A,B,\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}C.$ Sketch the parallelepiped and tetrahedron.

Let $\text{u},\text{v},\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}\text{w}$ be three-dimensional vectors and $c$ be a real number. Prove the following properties of the cross product.

1. $\text{u}\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}\text{u}=0$
2. $\text{u}\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}\left(\text{v}+\text{w}\right)=\left(\text{u}\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}\text{v}\right)+\left(\text{u}\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}\text{w}\right)$
3. $c\left(\text{u}\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}\text{v}\right)=\left(c\text{u}\right)\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}\text{v}=\text{u}\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}\left(c\text{v}\right)$
4. $\text{u}·\left(\text{u}\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}\text{v}\right)=0$

Show that vectors $\text{u}=⟨1,0,-8⟩,$ $\text{v}=⟨0,1,6⟩,$ and $\text{w}=⟨-1,9,3⟩$ satisfy the following properties of the cross product.

1. $\text{u}\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}\text{u}=0$
2. $\text{u}\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}\left(\text{v}+\text{w}\right)=\left(\text{u}\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}\text{v}\right)+\left(\text{u}\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}\text{w}\right)$
3. $c\left(\text{u}\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}\text{v}\right)=\left(c\text{u}\right)\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}\text{v}=\text{u}\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}\left(c\text{v}\right)$
4. $\text{u}·\left(\text{u}\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}\text{v}\right)=0$

Nonzero vectors $\text{u},\text{v},\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}\text{w}$ are said to be linearly dependent if one of the vectors is a linear combination of the other two. For instance, there exist two nonzero real numbers $\alpha$ and $\beta$ such that $\text{w}=\alpha \text{u}+\beta \text{v}.$ Otherwise, the vectors are called linearly independent . Show that $\text{u},\text{v},\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}\text{w}$ are coplanar if and only if they are linear dependent.

Consider vectors $\text{u}=⟨1,4,-7⟩,$ $\text{v}=⟨2,-1,4⟩,$ $\text{w}=⟨0,-9,18⟩,$ and $\mathbf{\text{p}}=⟨0,-9,17⟩.$

1. Show that $\text{u},\text{v},\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}\text{w}$ are coplanar by using their triple scalar product
2. Show that $\text{u},\text{v},\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}\text{w}$ are coplanar, using the definition that there exist two nonzero real numbers $\alpha$ and $\beta$ such that $\text{w}=\alpha \text{u}+\beta \text{v}.$
3. Show that $\text{u},\text{v},\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}\text{p}$ are linearly independent—that is, none of the vectors is a linear combination of the other two.

Consider points $A\left(0,0,2\right),$ $B\left(1,0,2\right),$ $C\left(1,1,2\right),$ and $D\left(0,1,2\right).$ Are vectors $\stackrel{\to }{AB},$ $\stackrel{\to }{AC},$ and $\stackrel{\to }{AD}$ linearly dependent (that is, one of the vectors is a linear combination of the other two)?

Yes, $\stackrel{\to }{AD}=\alpha \stackrel{\to }{AB}+\beta \stackrel{\to }{AC},$ where $\alpha =-1$ and $\beta =1.$

Show that vectors $\text{i}+\text{j},$ $\text{i}-\text{j},$ and $\text{i}+\text{j}+\text{k}$ are linearly independent—that is, there exist two nonzero real numbers $\alpha$ and $\beta$ such that $\text{i}+\text{j}+\text{k}=\alpha \left(\text{i}+\text{j}\right)+\beta \left(\text{i}-\text{j}\right).$

Let $\text{u}=⟨{u}_{1},{u}_{2}⟩$ and $\text{v}=⟨{v}_{1},{v}_{2}⟩$ be two-dimensional vectors. The cross product of vectors $\text{u}$ and $\text{v}$ is not defined. However, if the vectors are regarded as the three-dimensional vectors $\stackrel{˜}{\text{u}}=⟨{u}_{1},{u}_{2},0⟩$ and $\stackrel{˜}{\text{v}}=⟨{v}_{1},{v}_{2},0⟩,$ respectively, then, in this case, we can define the cross product of $\stackrel{˜}{\text{u}}$ and $\stackrel{˜}{\text{v}}.$ In particular, in determinant notation, the cross product of $\stackrel{˜}{\text{u}}$ and $\stackrel{˜}{\text{v}}$ is given by

$\stackrel{˜}{\text{u}}\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}\stackrel{˜}{\text{v}}=|\begin{array}{ccc}\text{i}\hfill & \text{j}\hfill & \text{k}\hfill \\ {u}_{1}\hfill & {u}_{2}\hfill & 0\hfill \\ {v}_{1}\hfill & {v}_{2}\hfill & 0\hfill \end{array}|.$

Use this result to compute $\left(\text{i}\text{cos}\phantom{\rule{0.2em}{0ex}}\theta +\text{j}\text{sin}\phantom{\rule{0.2em}{0ex}}\theta \right)\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}\left(\text{i}sin\theta -\text{j}cos\theta \right),$ where $\theta$ is a real number.

$\text{−}\text{k}$

Consider points $P\left(2,1\right),$ $Q\left(4,2\right),$ and $R\left(1,2\right).$

1. Find the area of triangle $P,Q,\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}R.$
2. Determine the distance from point $R$ to the line passing through $P\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}Q.$

Determine a vector of magnitude $10$ perpendicular to the plane passing through the x -axis and point $P\left(1,2,4\right).$

$⟨0,\text{±}4\sqrt{5},2\sqrt{5}⟩$

Determine a unit vector perpendicular to the plane passing through the z -axis and point $A\left(3,1,-2\right).$

Consider $\text{u}$ and $\text{v}$ two three-dimensional vectors. If the magnitude of the cross product vector $\text{u}\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}\text{v}$ is $k$ times larger than the magnitude of vector $\text{u},$ show that the magnitude of $\text{v}$ is greater than or equal to $k,$ where $k$ is a natural number.

[T] Assume that the magnitudes of two nonzero vectors $\text{u}$ and $\text{v}$ are known. The function $f\left(\theta \right)=‖\text{u}‖‖\text{v}‖\text{sin}\phantom{\rule{0.2em}{0ex}}\theta$ defines the magnitude of the cross product vector $\text{u}\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}\mathbf{\text{v}},$ where $\theta \in \left[0,\pi \right]$ is the angle between $\text{u}\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}\text{v}.$

1. Graph the function $f.$
2. Find the absolute minimum and maximum of function $f.$ Interpret the results.
3. If $‖\text{u}‖=5$ and $‖\text{v}‖=2,$ find the angle between $\text{u}\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}\text{v}$ if the magnitude of their cross product vector is equal to $9.$

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