2.4 The cross product  (Page 8/16)

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

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

$\text{u}=\overrightarrow{AB},$ $\text{v}=\overrightarrow{AC},$ where $A(1,0,1),$ $B(1,\mathrm{-1},3),$ and $C(0,0,5)$

$\text{u}=\overrightarrow{OP},$ $\text{v}=\overrightarrow{PQ},$ where $P(\mathrm{-1},1,0)$ and $Q(0,2,1)$

Determine the real number $\alpha$ such that $\text{u}\times \text{v}$ and $\text{i}$ are orthogonal, where $\text{u}=3\text{i}+\text{j}-5\text{k}$ and $\text{v}=4\text{i}-2\text{j}+\alpha \text{k}.$

Show that $\text{u}\times \text{v}$ 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 $\text{u}\times \text{v}$ is orthogonal to $\text{u}+\text{v}$ and $\text{u}-\text{v},$ where $\text{u}$ and $\text{v}$ are nonzero vectors.

Show that $\text{v}\times \text{u}$ 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 $\left|\begin{array}{ccc}\text{i}\hfill & \hfill \text{j}& \hfill \text{k}\\ 1\hfill & \hfill \mathrm{-1}& \hfill 7\\ 2\hfill & \hfill 0& \hfill 3\end{array}\right|.$

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

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

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

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

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

[T] Use the cross product $\text{u}\times \text{v}$ 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.

[T] Use the cross product $\text{u}\times \text{v}$ 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: ${\Vert \text{u}\times \text{v}\Vert }^2={\Vert \text{u}\Vert }^2{\Vert \text{v}\Vert }^2-{\left(\text{u}·\text{v}\right)}^2.$

Verify Lagrange’s identity ${\Vert \text{u}\times \text{v}\Vert }^2={\Vert \text{u}\Vert }^2{\Vert \text{v}\Vert }^2-{\left(\text{u}·\text{v}\right)}^2$ 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 $\text{u}\times \text{v}=0.$

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 $\overrightarrow{AB}$ and $\overrightarrow{AC}$ are collinear, where $A\left(4,1,0\right),$ $B\left(6,5,\mathrm{-2}\right),$ and $C\left(5,3,\mathrm{-1}\right).$

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

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

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

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

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

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

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

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

Calculate the triple scalar products $\text{v}·(\text{u}\times \mathbf{\text{w}})$ and $\text{w}·(\text{u}\times \text{v}),$ where $\text{u}=⟨1,1,1⟩,$ $\text{v}=⟨7,6,9⟩,$ and $\text{w}=⟨4,2,7⟩.$

Calculate the triple scalar products $\text{w}·(\text{v}\times \text{u})$ and $\text{u}·(\mathbf{\text{w}}\times \text{v}),$ where $\text{u}=⟨4,2,\mathrm{-1}⟩,$ $\text{v}=⟨2,5,\mathrm{-3}⟩,$ and $\text{w}=⟨9,5,\mathrm{-10}⟩.$

Find vectors $\text{a},\text{b},\text{and}\text{c}$ with a triple scalar product given by the determinant

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

The triple scalar product of vectors $\text{a},\text{b},\text{and}\text{c}$ is given by the determinant

$\left|\begin{array}{ccc}\hfill 0& \hfill \mathrm{-2}& \hfill 1\\ \hfill 0& \hfill 1& \hfill 4\\ \hfill 1& \hfill \mathrm{-3}& \hfill 7\end{array}\right|.$ 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.