외적

[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 $\text{u}=-\text{i}+3\text{j}+\text{k}$ 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: ${\| \text{u}\,\times \,\text{v}\| }^{2}={\| \text{u}\| }^{2}{\| \text{v}\| }^{2}-{(\text{u}\cdot \text{v})}^{2}.$

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 $\text{v}.$

Verify Lagrange’s identity ${\| \text{u}\,\times \,\text{v}\| }^{2}={\| \text{u}\| }^{2}{\| \text{v}\| }^{2}-{(\text{u}\cdot \text{v})}^{2}$ for vectors $\text{u}=-\text{i}+\text{j}-2\text{k}$ 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 $\vec{AB}$ and $\vec{AC}$ are collinear, where $A(4,1,0),$ $B(6,5,-2),$ and $C(5,3,-1).$

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

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

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

a. Find the area of parallelogram $ABCD$ with adjacent sides $\vec{AB}$ and $\vec{AC}.$ b. Find the area of triangle $ABC.$ c. Find the distance from point $A$ to line $BC.$

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

a. Find the area of parallelogram $ABCD$ with adjacent sides $\vec{AB}$ and $\vec{AC}.$ b. Find the area of triangle $ABC.$ c. Find the distance from point $B$ to line $AC.$

In the following exercises, vectors $\text{u},\text{v},\text{and}\,\text{w}$ are given.

a. Find the triple scalar product $\text{u}\cdot (\text{v}\,\times \,\text{w}).$ b. Find the volume of the parallelepiped with the adjacent edges $\text{u},\text{v},\text{and}\,\text{w}.$

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

In the following exercises, vectors $\text{u},\text{v},\text{and}\,\text{w}$ are given.

a. Find the triple scalar product $\text{u}\cdot (\text{v}\,\times \,\text{w}).$ b. Find the volume of the parallelepiped with the adjacent edges $\text{u},\text{v},\text{and}\,\text{w}.$

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

Calculate the triple scalar products $\text{v}\cdot (\text{u}\,\times \,\text{w})$ and $\text{w}\cdot (\text{u}\,\times \,\text{v}),$ where $\text{u}=\langle 1,1,1\rangle ,$ $\text{v}=\langle 7,6,9\rangle ,$ and $\text{w}=\langle 4,2,7\rangle .$

Calculate the triple scalar products $\text{w}\cdot (\text{v}\,\times \,\text{u})$ and $\text{u}\cdot (\text{w}\,\times \,\text{v}),$ where $\text{u}=\langle 4,2,-1\rangle ,$ $\text{v}=\langle 2,5,-3\rangle ,$ and $\text{w}=\langle 9,5,-10\rangle .$

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

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

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

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

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

a. Find the real number $\alpha >0$ such that the volume of the parallelepiped is $3$ units³. b. For $\alpha =1,$ find the height $h$ from vertex $C$ of the parallelepiped to the plane formed by the edges $OA$ and $OB$.

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

a. Determine the volume of the parallelepiped with adjacent sides $\vec{OA},$ $\vec{OB},$ and $\vec{OC}.$ b. Find the volume of the tetrahedron with vertices $O,A,B,\,\text{and}\,C.$ (Hint: The volume of the tetrahedron is $1/6$ of the volume of the parallelepiped.) c. Find the distance from the origin to the plane determined by $A,B,\,\text{and}\,C.$ Sketch the parallelepiped and tetrahedron.

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

a. $\text{u}\,\times \,\text{u}=0$ b. $\text{u}\,\times \,(\text{v}+\text{w})=(\text{u}\,\times \,\text{v})+(\text{u}\,\times \,\text{w})$ c. $c(\text{u}\,\times \,\text{v})=(c\text{u})\,\times \,\text{v}=\text{u}\,\times \,(c\text{v})$ d. $\text{u}\cdot (\text{u}\,\times \,\text{v})=0$

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

a. $\text{u}\,\times \,\text{u}=0$ b. $\text{u}\,\times \,(\text{v}+\text{w})=(\text{u}\,\times \,\text{v})+(\text{u}\,\times \,\text{w})$ c. $c(\text{u}\,\times \,\text{v})=(c\text{u})\,\times \,\text{v}=\text{u}\,\times \,(c\text{v})$ d. $\text{u}\cdot (\text{u}\,\times \,\text{v})=0$