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

$x=1+t,y=3+t,z=5+4t,$ $t\in ℝ$

$\text{−}x=y+1,z=2$

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

$x=\text{−}y=\text{−}z.$

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

$x=\mathrm{-1}-t,y=\text{−}t,z=2,$ $t\in ℝ.$

$L_1:x=1+t,y=t,z=2+t,$ $t\in ℝ,$ $L_2:x-3=y-1=z-3$

$L_1:x=2,y=1,z=t,$ $L_2:x=1,y=1,z=2-3t,$ $t\in ℝ$

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

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

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

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

$x=10+t,y=2-2t,z=\mathrm{-3}+3t,$ $t\in ℝ.$

$L_1:x=y-1=\text{−}z$ and $L_2:x-2=\text{−}y=\frac{z}{2}$

$L_1:x=2t,y=0,z=3,$ $t\in ℝ$ and $L_2:x=0,y=8+s,z=7+s,$ $s\in ℝ$

$L_1:x=\mathrm{-1}+2t,y=1+3t,z=7t,$ $t\in ℝ$ and $L_2:x-1=\frac{2}{3}\left(y-4\right)=\frac{2}{7}z-2$

$L_1:3x=y+1=2z$ 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 $A(1,1,1).$

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 $A.$

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.

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

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

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

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

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

$3x+4y-12=0$

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

$x+z=0$

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 $\overrightarrow{PQ}$ and $\text{n}$ are orthogonal.

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 $P(\mathrm{-2},1,3)$ that is perpendicular to the plane of equation $2x-3y+z=7.$