벡터와 공간 전체 문제

A baseball player throws a baseball at an angle of $30^\circ$ with the horizontal. If the initial speed of the ball is $100$ mph, find the horizontal and vertical components of the initial velocity vector of the baseball. (Round to two decimal places.)

Let $\text{r}(t)=\langle x(t),y(t),z(t)\rangle$ be the position vector of a particle at the time $t\in [0,T],$ where $x,y,$ and $z$ are smooth functions on $[0,T].$ The instantaneous velocity of the particle at time $t$ is defined by vector $\text{v}(t)=\langle x'(t),y'(t),z'(t)\rangle ,$ with components that are the derivatives with respect to $t,$ of the functions x, y, and z, respectively. The magnitude $\| \text{v}(t)\|$ of the instantaneous velocity vector is called the speed of the particle at time t. Vector $\text{a}(t)=\langle {x}^{''}(t),{y}^{''}(t),{z}^{''}(t)\rangle ,$ with components that are the second derivatives with respect to $t,$ of the functions $x,y,$ and $z,$ respectively, gives the acceleration of the particle at time $t.$ 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 is expressed in seconds.

a. Find the instantaneous velocity, speed, and acceleration of the particle after the first second. Round your answer to two decimal places. b. Use a CAS to visualize the path of the particle—that is, the set of all points of coordinates $(\cos \,t,\sin \,t,2t),$ where $t\in [0,30].$

[T] Let $\text{r}(t)=\langle t,2{t}^{2},4{t}^{2}\rangle$ be the position vector of a particle at time $t$ (in seconds), where $t\in [0,10]$ (here the components of $\text{r}$ are expressed in centimeters).

a. Find the instantaneous velocity, speed, and acceleration of the particle after the first two seconds. Round your answer to two decimal places. b. Use a CAS to visualize the path of the particle defined by the points $(t,2{t}^{2},4{t}^{2}),$ where $t\in [0,60].$

[T] Find the vectors that join the center of a clock to the hours 1:00, 2:00, and 3:00. Assume the clock is circular with a radius of 1 unit.

Find the work done by force $\text{F}=\langle 5,6,-2\rangle$ (measured in Newtons) that moves a particle from point $P(3,-1,0)$ to point $Q(2,3,1)$ along a straight line (the distance is measured in meters).

[T] Two forces ${\text{F}}_{1}$ and ${\text{F}}_{2}$ are represented by vectors with initial points that are at the origin. The first force has a magnitude of 20 lb and the terminal point of the vector is point $P(1,1,0).$ The second force has a magnitude of 40 lb and the terminal point of its vector is point $Q(0,1,1).$ Let F be the resultant force of forces ${\text{F}}_{1}$ and ${\text{F}}_{2}.$

a. Find the magnitude of F. (Round the answer to one decimal place.) b. Find the direction angles of F. (Express the answer in degrees rounded to one decimal place.)

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.

[T] Consider $\text{r}(t)=\langle \sin \,t,\cos \,t,2t\rangle$ the position vector of a particle at time $t\in [0,3],$ where the components of r are expressed in centimeters and time is measured in seconds. Let $\vec{OP}$ be the position vector of the particle after $1$ sec.

a. Determine the velocity vector $\text{v}(1)$ of the particle after $1$ sec. b. Find the scalar equation of the plane that is perpendicular to $\text{v}(1)$ and passes through point $P.$ This plane is called the normal plane to the path of the particle at point $P.$ c. Use a CAS to visualize the path of the particle along with the velocity vector and normal plane at point $P.$

For the following exercises, find the area or volume of the given shapes.

The parallelogram spanned by vectors $\text{a}=\langle 1,13\rangle \,\text{and}\,\text{b}=\langle 3,21\rangle$