외적

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 $\text{w}=\langle 3,1,1\rangle$

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 $P(2,1),$ $Q(4,2),$ and $R(1,2).$

a. Find the area of triangle $P,Q,\,\text{and}\,R.$ b. Determine the distance from point $R$ to the line passing through $P\,\text{and}\,Q.$

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 $\text{u},\text{v},\text{and}\,\text{w}.$

Find all vectors $\text{w}=\langle {w}_{1},{w}_{2},{w}_{3}\rangle$ that satisfy the equation $\langle 1,1,1\rangle \,\times \,\text{w}=\langle -1,-1,2\rangle .$

[T] The force vector $\text{F}$ acting on a proton with an electric charge of $1.6\,\times \,{10}^{-19}\text{C}$ (in coulombs) moving in a magnetic field $\text{B}$ where the velocity vector $\text{v}$ is given by $\text{F}=1.6\,\times \,{10}^{-19}(\text{v}\,\times \,\text{B})$ (here, $\text{v}$ is expressed in meters per second, $\text{B}$ is in tesla [T], and $\text{F}$ is in newtons [N]). Find the force that acts on a proton that moves in the xy-plane at velocity $\text{v}={10}^{5}\text{i}+{10}^{5}\text{j}$ (in meters per second) in a magnetic field given by $\text{B}=0.3\text{j}.$

[T] The force vector $\text{F}$ acting on a proton with an electric charge of $1.6\,\times \,{10}^{-19}\text{C}$ moving in a magnetic field $\text{B}$ where the velocity vector v is given by $\text{F}=1.6\,\times \,{10}^{-19}(\text{v}\,\times \,\text{B})$ (here, $\text{v}$ is expressed in meters per second, $\text{B}$ in $\text{T},$ and $\text{F}$ in $\text{N}).$ If the magnitude of force $\text{F}$ acting on a proton is $5.9\,\times \,{10}^{-17}$ N and the proton is moving at the speed of 300 m/sec in magnetic field $\text{B}$ of magnitude 2.4 T, find the angle between velocity vector $\text{v}$ of the proton and magnetic field $\text{B}.$ Express the answer in degrees rounded to the nearest integer.

[T] Consider $\text{r}(t)=\langle \cos \,t,\sin \,t,2t\rangle$ the position vector of a particle at time $t\in [0,30],$ where the components of $\text{r}$ are expressed in centimeters and time in seconds. Let $\vec{OP}$ be the position vector of the particle after $1$ sec.

a. Determine unit vector $\text{B}(t)$ (called the binormal unit vector) that has the direction of cross product vector $\text{v}(t)\,\times \,\text{a}(t),$ where $\text{v}(t)$ and $\text{a}(t)$ are the instantaneous velocity vector and, respectively, the acceleration vector of the particle after $t$ seconds. b. Use a CAS to visualize vectors $\text{v}(1),$ $\text{a}(1),$ and $\text{B}(1)$ as vectors starting at point $P$ along with the path of the particle.

In Exercises 23–26, find the area of the parallelogram defined by the given vectors.

$\vec u = \left\langle 1,1,2\right\rangle$, $\vec v = \left\langle 2,0,3\right\rangle$

In Exercises 23–26, find the area of the parallelogram defined by the given vectors.

$\vec u = \left\langle -2,1,5\right\rangle$, $\vec v = \left\langle -1,3,1\right\rangle$

In Exercises 23–26, find the area of the parallelogram defined by the given vectors.

$\vec u = \left\langle 1,2\right\rangle$, $\vec v = \left\langle 2,1\right\rangle$

In Exercises 23–26, find the area of the parallelogram defined by the given vectors.

$\vec u = \left\langle 2,0\right\rangle$, $\vec v = \left\langle 0,3\right\rangle$

In Exercises 27–30, find the area of the triangle with the given vertices.

Vertices: $(0,0,0)$, $(1,3,-1)$ and $(2,1,1)$.

In Exercises 27–30, find the area of the triangle with the given vertices.

Vertices: $(5,2,-1)$, $(3,6,2)$ and $(1,0,4)$.

In Exercises 27–30, find the area of the triangle with the given vertices.

Vertices: $(1,1)$, $(1,3)$ and $(2,2)$.

In Exercises 27–30, find the area of the triangle with the given vertices.

Vertices: $(3,1)$, $(1,2)$ and $(4,3)$.