2.5 Equations of lines and planes in space  (Page 10/19)

Find symmetric equations of the line passing through point $P(2,5,4)$ that is perpendicular to the plane of equation $2x+3y-5z=0.$

Show that line $\frac{x-1}{2}=\frac{y+1}{3}=\frac{z-2}{4}$ is parallel to plane $x-2y+z=6.$

Find the real number $\alpha$ such that the line of parametric equations $x=t,y=2-t,z=3+t,$ $t\in ℝ$ is parallel to the plane of equation $\alpha x+5y+z-10=0.$

$P(1,1,1),Q(2,4,3),$ and $R(\mathrm{-1},\mathrm{-2},\mathrm{-1})$

$P\left(\mathrm{-2},1,4\right),Q\left(3,1,3\right),$ and $R\left(\mathrm{-2},1,0\right)$

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

1. Show that the planes intersect.
2. 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 $\text{−}y+z-2=0$ and $x-y=0.$

1. Show that the planes intersect.
2. Find parametric equations of the line passing through point $P(\mathrm{-8},0,2)$ that is parallel to the line of intersection of the planes.

Find the scalar equation of the plane that passes through point $P(\mathrm{-1},2,1)$ 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 $\text{−}x+y+2=0$ and $z-3=0.$

Determine whether the line of parametric equations $x=1+2t,y=\mathrm{-2}t,z=2+t,$ $t\in ℝ$ 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 $x=5,y=4-t,z=2t,$ $t\in ℝ$ intersects the plane with equation $2x-y+z=5.$ If it does intersect, find the point of intersection.

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

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

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

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

$x-5y-z=1,$ $5x-25y-5z=\mathrm{-3}$

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

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

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

Consider point $C\left(\mathrm{-3},2,4\right)$ and the plane of equation $2x+4y-3z=8.$

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

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

1. Find the equation of the sphere with center $C$ at the origin that is tangent to the given plane.
2. 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] John allocates $d$ dollars to consume monthly three goods of prices $a,b,\text{and}c.$ In this context, the budget equation is defined as $ax+by+cz=d,$ where $x\ge 0,y\ge 0,$ and $z\ge 0$ represent the number of items bought from each of the goods. The budget set is given by $\left\{\left(x,y,z\right)|ax+by+cz\le d,x\ge 0,y\ge 0,z\ge 0\right\},$ and the budget plane is the part of the plane of equation $ax+by+cz=d$ for which $x\ge 0,y\ge 0,$ and $z\ge 0.$ Consider $a=\text{\$}8,$ $b=\text{\$}5,$ $c=\text{\$}10,$ and $d=\text{\$}500.$

1. Use a CAS to graph the budget set and budget plane.
2. For $z=25,$ find the new budget equation and graph the budget set in the same system of coordinates.

[T] Consider $\text{r}(t)=⟨\text{sin}t,\text{cos}t,2t⟩$ 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 $\overrightarrow{OP}$ be the position vector of the particle after $1$ sec.

1. Determine the velocity vector $\text{v}(1)$ of the particle after $1$ sec.
2. 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 $P.$
3. Use a CAS to visualize the path of the particle along with the velocity vector and normal plane at point $P.$

[T] A solar panel is mounted on the roof of a house. The panel may be regarded as positioned at the points of coordinates (in meters) $A(8,0,0),$ $B(8,18,0),$ $C(0,18,8),$ and $D(0,0,8)$ (see the following figure).

1. Find the general form of the equation of the plane that contains the solar panel by using points $A,B,\text{and}C,$ and show that its normal vector is equivalent to $\overrightarrow{AB}\times \overrightarrow{AD}.$
2. Find parametric equations of line $L_1$ that passes through the center of the solar panel and has direction vector $\text{s}=\frac{1}{\sqrt{3}}\text{i}+\frac{1}{\sqrt{3}}\text{j}+\frac{1}{\sqrt{3}}\text{k},$ which points toward the position of the Sun at a particular time of day.
3. Find symmetric equations of line $L_2$ that passes through the center of the solar panel and is perpendicular to it.
4. Determine the angle of elevation of the Sun above the solar panel by using the angle between lines $L_1$ and $L_2.$