벡터와 공간 전체 문제

For the following exercises, points $P,Q,\,\text{and}\,R$ are given.

a. Find the general equation of the plane passing through $P,Q,\,\text{and}\,R.$ b. Write the vector equation $\text{n}\cdot \vec{PS}=0$ of the plane at a., where $S(x,y,z)$ is an arbitrary point of the plane. c. Find parametric equations of the line passing through the origin that is perpendicular to the plane passing through $P,Q,\,\text{and}\,R.$

$P(1,1,1),Q(2,4,3),$ and $R(-1,-2,-1)$

For the following exercises, points $P,Q,\,\text{and}\,R$ are given.

a. Find the general equation of the plane passing through $P,Q,\,\text{and}\,R.$ b. Write the vector equation $\text{n}\cdot \vec{PS}=0$ of the plane at a., where $S(x,y,z)$ is an arbitrary point of the plane. c. Find parametric equations of the line passing through the origin that is perpendicular to the plane passing through $P,Q,\,\text{and}\,R.$

$P(-2,1,4),Q(3,1,3),$ and $R(-2,1,0)$

Consider the planes of equations $x+y+z=1$ and $x+z=0.$

a. Show that the planes intersect. b. Find symmetric equations of the line passing through point $P(1,4,6)$ that is parallel to the line of intersection of the planes.

Consider the planes of equations $-y+z-2=0$ and $x-y=0.$

a. Show that the planes intersect. b. Find parametric equations of the line passing through point $P(-8,0,2)$ that is parallel to the line of intersection of the planes.

Find the scalar equation of the plane that passes through point $P(-1,2,1)$ and is perpendicular to the line of intersection of planes $x+y-z-2=0$ and $2x-y+3z-1=0.$

Find the general equation of the plane that passes through the origin and is perpendicular to the line of intersection of planes $-x+y+2=0$ and $z-3=0.$

Determine whether the line of parametric equations $x=1+2t,y=-2t,z=2+t,$ $t\in \mathbb{R}$ intersects the plane with equation $3x+4y+6z-7=0.$ If it does intersect, find the point of intersection.

Determine whether the line of parametric equations $x=5,y=4-t,z=2t,$ $t\in \mathbb{R}$ intersects the plane with equation $2x-y+z=5.$ If it does intersect, find the point of intersection.

Find the distance from point $P(1,5,-4)$ to the plane of equation $3x-y+2z-6=0.$

Find the distance from point $P(1,-2,3)$ to the plane of equation $(x-3)+2(y+1)-4z=0.$

For the following exercises, the equations of two planes are given.

a. Determine whether the planes are parallel, orthogonal, or neither. b. If the planes are neither parallel nor orthogonal, then find the measure of the angle between the planes. Express the answer in degrees rounded to the nearest integer.

[T] $x+y+z=0,$ $2x-y+z-7=0$

For the following exercises, the equations of two planes are given.

a. Determine whether the planes are parallel, orthogonal, or neither. b. If the planes are neither parallel nor orthogonal, then find the measure of the angle between the planes. Express the answer in degrees rounded to the nearest integer.

$5x-3y+z=4,$ $x+4y+7z=1$

For the following exercises, the equations of two planes are given.

a. Determine whether the planes are parallel, orthogonal, or neither. b. If the planes are neither parallel nor orthogonal, then find the measure of the angle between the planes. Express the answer in degrees rounded to the nearest integer.

$x-5y-z=1,$ $5x-25y-5z=-3$

For the following exercises, the equations of two planes are given.

a. Determine whether the planes are parallel, orthogonal, or neither. b. If the planes are neither parallel nor orthogonal, then find the measure of the angle between the planes. Express the answer in degrees rounded to the nearest integer.

[T] $x-3y+6z=4,$ $5x+y-z=4$

Show that the lines of equations $x=t,y=1+t,z=2+t,$ $t\in \mathbb{R},$ and $\frac{x}{2}=\frac{y-1}{3}=z-3$ are skew, and find the distance between them.

Show that the lines of equations $x=-1+t,y=-2+t,z=3t,$ $t\in \mathbb{R},$ and $x=5+s,y=-8+2s,z=7s,$ $s\in \mathbb{R}$ are skew, and find the distance between them.

Consider point $C(-3,2,4)$ and the plane of equation $2x+4y-3z=8.$

a. Find the radius of the sphere with center $C$ tangent to the given plane. b. Find point P of tangency.

Consider the plane of equation $x-y-z-8=0.$

a. Find an equation of the sphere with center $C$ at the origin that is tangent to the given plane. b. Find parametric equations of the line passing through the origin and the point of tangency.

Two children are playing with a ball. The girl throws the ball to the boy. The ball travels in

the air, curves $3$ ft to the right, and falls $5$ ft away from the girl (see the following figure). If the plane that contains the trajectory of the ball is perpendicular to the ground, find its equation.

[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 $P.$