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$x=1+t,y=3+t,z=5+4t,$ $t\in \mathbb{R}$
a. $P(1,3,5),$ $v=\u27e81,1,4\u27e9;$ b. $\sqrt{3}$
$\text{\u2212}x=y+1,z=2$
Find the distance between point $A\left(\mathrm{-3},1,1\right)$ and the line of symmetric equations
$x=\text{\u2212}y=\text{\u2212}z.$
$\frac{2\sqrt{2}}{\sqrt{3}}$
Find the distance between point $A\left(4,2,5\right)$ and the line of parametric equations
$x=\mathrm{-1}-t,y=\text{\u2212}t,z=2,$ $t\in \mathbb{R}.$
For the following exercises, lines ${L}_{1}$ and ${L}_{2}$ are given.
${L}_{1}:x=1+t,y=t,z=2+t,$ $t\in \mathbb{R},$ ${L}_{2}:x-3=y-1=z-3$
a. Parallel; b. $\frac{\sqrt{2}}{\sqrt{3}}$
${L}_{1}:x=2,y=1,z=t,$ ${L}_{2}:x=1,y=1,z=2-3t,$ $t\in \mathbb{R}$
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 $x=3t,y=3+8t,z=\mathrm{-7}+6t,$ $t\in \mathbb{R}.$
Are the lines of equations $x=\mathrm{-2}+2t,y=\mathrm{-6},z=2+6t$ and $x=\mathrm{-1}+t,y=1+t,z=t,$ $t\in \mathbb{R},$ perpendicular to each other?
Find the point of intersection of the lines of equations $x=\mathrm{-2}y=3z$ and $x=\mathrm{-5}-t,y=\mathrm{-1}+t,z=t-11,$ $t\in \mathbb{R}.$
$\left(\mathrm{-12},6,\mathrm{-4}\right)$
Find the intersection point of the x -axis with the line of parametric equations
$x=10+t,y=2-2t,z=\mathrm{-3}+3t,$ $t\in \mathbb{R}.$
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.
${L}_{1}:x=y-1=\text{\u2212}z$ and ${L}_{2}:x-2=\text{\u2212}y=\frac{z}{2}$
The lines are skew.
${L}_{1}:x=2t,y=0,z=3,$ $t\in \mathbb{R}$ and ${L}_{2}:x=0,y=8+s,z=7+s,$ $s\in \mathbb{R}$
${L}_{1}:x=\mathrm{-1}+2t,y=1+3t,z=7t,$ $t\in \mathbb{R}$ 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 ${L}_{2}:x=6+2t,y=17+6t,z=9+3t,$ $t\in \mathbb{R}$
Consider line $L$ of symmetric equations $x-2=\text{\u2212}y=\frac{z}{2}$ and point $A(1,1,1).$
a. $x=1+t,y=1-t,z=1+2t,$ $t\in \mathbb{R};$ 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 $(2,0,0)$ that belongs to $L$ is a line that intersects; $L:\frac{x-1}{\mathrm{-1}}=y-1=z-1$
Consider line $L$ of parametric equations $x=t,y=2t,z=3,$ $t\in \mathbb{R}.$
For the following exercises, point $P$ and vector $\text{n}$ are given.
$P(0,0,0),$ $\text{n}=3\text{i}-2\text{j}+4\text{k}$
a. $3x-2y+4z=0;$ b. $3x-2y+4z=0$
$P\left(3,2,2\right),$ $\text{n}=2\text{i}+3\text{j}-\text{k}$
$P\left(1,2,3\right),$ $\text{n}=\u27e81,2,3\u27e9$
a. $\left(x-1\right)+2\left(y-2\right)+3\left(z-3\right)=0;$ b. $x+2y+3z-14=0$
$P(0,0,0),$ $\text{n}=\u27e8\mathrm{-3},2,\mathrm{-1}\u27e9$
For the following exercises, the equation of a plane is given.
[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.
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.
$\left(3,0,0\right)$
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 $P(\mathrm{-2},1,3)$ that is perpendicular to the plane of equation $2x-3y+z=7.$
$x=\mathrm{-2}+2t,y=1-3t,z=3+t,$ $t\in \mathbb{R}$
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