# 2.5 Equations of lines and planes in space  (Page 9/19)

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$x=1+t,y=3+t,z=5+4t,$ $t\in ℝ$

a. $P\left(1,3,5\right),$ $v=⟨1,1,4⟩;$ b. $\sqrt{3}$

$\text{−}x=y+1,z=2$

Find the distance between point $A\left(-3,1,1\right)$ and the line of symmetric equations

$x=\text{−}y=\text{−}z.$

$\frac{2\sqrt{2}}{\sqrt{3}}$

Find the distance between point $A\left(4,2,5\right)$ and the line of parametric equations

$x=-1-t,y=\text{−}t,z=2,$ $t\in ℝ.$

For the following exercises, lines ${L}_{1}$ and ${L}_{2}$ are given.

1. Verify whether lines ${L}_{1}$ and ${L}_{2}$ are parallel.
2. If the lines ${L}_{1}$ and ${L}_{2}$ are parallel, then find the distance between them.

${L}_{1}:x=1+t,y=t,z=2+t,$ $t\in ℝ,$ ${L}_{2}:x-3=y-1=z-3$

a. Parallel; b. $\frac{\sqrt{2}}{\sqrt{3}}$

${L}_{1}:x=2,y=1,z=t,$ ${L}_{2}:x=1,y=1,z=2-3t,$ $t\in ℝ$

Show that the line passing through points $P\left(3,1,0\right)$ and $Q\left(1,4,-3\right)$ is perpendicular to the line with equation $x=3t,y=3+8t,z=-7+6t,$ $t\in ℝ.$

Are the lines of equations $x=-2+2t,y=-6,z=2+6t$ and $x=-1+t,y=1+t,z=t,$ $t\in ℝ,$ perpendicular to each other?

Find the point of intersection of the lines of equations $x=-2y=3z$ and $x=-5-t,y=-1+t,z=t-11,$ $t\in ℝ.$

$\left(-12,6,-4\right)$

Find the intersection point of the x -axis with the line of parametric equations

$x=10+t,y=2-2t,z=-3+3t,$ $t\in ℝ.$

For the following exercises, lines ${L}_{1}$ and ${L}_{2}$ are given. Determine whether the lines are equal, parallel but not equal, skew, or intersecting.

${L}_{1}:x=y-1=\text{−}z$ and ${L}_{2}:x-2=\text{−}y=\frac{z}{2}$

The lines are skew.

${L}_{1}:x=2t,y=0,z=3,$ $t\in ℝ$ and ${L}_{2}:x=0,y=8+s,z=7+s,$ $s\in ℝ$

${L}_{1}:x=-1+2t,y=1+3t,z=7t,$ $t\in ℝ$ and ${L}_{2}:x-1=\frac{2}{3}\left(y-4\right)=\frac{2}{7}z-2$

The lines are equal.

${L}_{1}:3x=y+1=2z$ and ${L}_{2}:x=6+2t,y=17+6t,z=9+3t,$ $t\in ℝ$

Consider line $L$ of symmetric equations $x-2=\text{−}y=\frac{z}{2}$ and point $A\left(1,1,1\right).$

1. Find parametric equations for a line parallel to $L$ that passes through point $A.$
2. Find symmetric equations of a line skew to $L$ and that passes through point $A.$
3. Find symmetric equations of a line that intersects $L$ and passes through point $A.$

a. $x=1+t,y=1-t,z=1+2t,$ $t\in ℝ;$ b. For instance, the line passing through $A$ with direction vector $\mathbf{\text{j}}:x=1,z=1;$ c. For instance, the line passing through $A$ and point $\left(2,0,0\right)$ that belongs to $L$ is a line that intersects; $L:\frac{x-1}{-1}=y-1=z-1$

Consider line $L$ of parametric equations $x=t,y=2t,z=3,$ $t\in ℝ.$

1. Find parametric equations for a line parallel to $L$ that passes through the origin.
2. Find parametric equations of a line skew to $L$ that passes through the origin.
3. Find symmetric equations of a line that intersects $L$ and passes through the origin.

For the following exercises, point $P$ and vector $\text{n}$ are given.

1. Find the scalar equation of the plane that passes through $P$ and has normal vector $\text{n}.$
2. Find the general form of the equation of the plane that passes through $P$ and has normal vector $\text{n}.$

$P\left(0,0,0\right),$ $\text{n}=3\text{i}-2\text{j}+4\text{k}$

a. $3x-2y+4z=0;$ b. $3x-2y+4z=0$

$P\left(3,2,2\right),$ $\text{n}=2\text{i}+3\text{j}-\text{k}$

$P\left(1,2,3\right),$ $\text{n}=⟨1,2,3⟩$

a. $\left(x-1\right)+2\left(y-2\right)+3\left(z-3\right)=0;$ b. $x+2y+3z-14=0$

$P\left(0,0,0\right),$ $\text{n}=⟨-3,2,-1⟩$

For the following exercises, the equation of a plane is given.

1. Find normal vector $\text{n}$ to the plane. Express $\text{n}$ using standard unit vectors.
2. Find the intersections of the plane with the axes of coordinates.
3. Sketch the plane.

[T] $4x+5y+10z-20=0$

a. $\mathbf{\text{n}}=4\text{i}+5\mathbf{\text{j}}+10\mathbf{\text{k}};$ b. $\left(5,0,0\right),$ $\left(0,4,0\right),$ and $\left(0,0,2\right);$
c.

$3x+4y-12=0$

$3x-2y+4z=0$

a. $\text{n}=3\text{i}-2\text{j}+4\text{k};$ b. $\left(0,0,0\right);$
c.

$x+z=0$

Given point $P\left(1,2,3\right)$ and vector $\text{n}=\text{i}+\text{j},$ find point $Q$ on the x -axis such that $\stackrel{\to }{PQ}$ and $\text{n}$ are orthogonal.

$\left(3,0,0\right)$

Show there is no plane perpendicular to $\text{n}=\text{i}+\text{j}$ that passes through points $P\left(1,2,3\right)$ and $Q\left(2,3,4\right).$

Find parametric equations of the line passing through point $P\left(-2,1,3\right)$ that is perpendicular to the plane of equation $2x-3y+z=7.$

$x=-2+2t,y=1-3t,z=3+t,$ $t\in ℝ$

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