Equations Of Lines And Planes

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1. Find a vector equation and parametric equations for the line through the point $\left(-2,4,10\right)$ and parallel to the vector $⟨3,1,-8⟩$.
   $\mathbf{r}=\left(-2\mathbf{i}+4\mathbf{j}+10\mathbf{k}\right)+t\left(3\mathbf{i}+\mathbf{j}-8\mathbf{k}\right)$
   $x=-2+3t$, $y=4+t$, $z=10-8t$
2. Find a vector equation and parametric equations for the line through the origin and parallel to the line $x=2t$, $y=1-t$, $z=4+3t$.
3. Find parametric equations and symmetric equations for the line through the points $\left(6,1,-3\right)$ and $\left(2,4,5\right)$.
4. Find a vector equation and parametric equations for the line through the point $\left(1,0,6\right)$ and perpendicular to the plane $x+3y+z=5$.
   $\mathbf{r}=\left(\mathbf{i}+6\mathbf{k}\right)+t\left(\mathbf{i}+3\mathbf{j}+\mathbf{k}\right)$,
   $x=1+t$, $y=3t$, and $z=6+t$.
5. Is the line through $\left(-4,-6,1\right)$ and $\left(-2,0,-3\right)$ parallel to the line through $\left(10,18,4\right)$ and $\left(5,3,14\right)$?
   Yes.
6. 1. Find symmetric equations for the line that passes through the point $\left(0,2,-1\right)$ and is parallel to the line with parametric equations $x=1+2t$, $y=3t$, and $z=5-7t$.
   2. Find the points in which the required line in part (a) intersects the coordinate planes.
   1. ${\displaystyle \frac{x}{2}=\frac{y-2}{3}=\frac{z+1}{-7}}$.
   2. $\left(-\frac{2}{7},\frac{11}{7},0\right)$, $\left(-\frac{4}{3},0,\frac{11}{3}\right)$, and $\left(0,2,-1\right)$.
7. Find parametric equations for the line segment from $\left(10,3,1\right)$ to $\left(5,6,-3\right)$.

8. Determine whether the lines $L_1$ and $L_2$ are parallel, skew, or intersecting. Find the point of intersection, if any.

   $L_1$: $x=-6t$, $y=1+9t$, $z=-3t$,

   $L_2$: $x=1+2s$, $y=4-3s$, $z=s$.

   Parallel.

9. Determine whether the lines $L_1$ and $L_2$ are parallel, skew, or intersecting. Find the point of intersection, if any.

   $L_1$: $x=1+2t$, $y=3t$, $z=2-t$,

   $L_2$: $x=-1+s$, $y=4+s$, $z=1+3s$.

   Skew.

10. Determine whether the lines $L_1$ and $L_2$ are parallel, skew, or intersecting. Find the point of intersection, if any.

    $L_1$: ${\displaystyle x=\frac{y-1}{2}=\frac{z-2}{3}}$,

    $L_2$: ${\displaystyle \frac{x-3}{-4}=\frac{y-2}{-3}=\frac{z-1}{2}}$.

    Skew.

11. Determine whether the lines $L_1$ and $L_2$ are parallel, skew, or intersecting. Find the point of intersection, if any.

    $L_1$: ${\displaystyle \frac{x-1}{2}=\frac{y-3}{2}=\frac{z-2}{-1}}$,

    $L_2$: ${\displaystyle x-2=\frac{y-6}{-1}=\frac{z+2}{3}}$.

12. Find an equation of the plane through the point $\left(4,0,-3\right)$ and with normal vector $\mathbf{j}+2\mathbf{k}$.
13. Find an equation of the plane through the origin and parallel to the plane $2x-y+3z=1$.
    $2x-y+3z=0$.
14. Find an equation of the plane through the origin and the points $\left(2,-4,6\right)$ and $\left(5,1,3\right)$.
15. Where does the line through $\left(1,0,1\right)$ and $\left(4,-2,2\right)$ intersect the plane $x+y+z=6$?
16. Determine whether the planes $x+y+z=1$ and $x-y+z=1$ are parallel, perpendicular, or neither. If neither, find the angle between them.
    Neither, the angle is approx. $70.5°$.
17. Determine whether the planes $x+2y+2z=1$ and $2x-y+2z=1$ are parallel, perpendicular, or neither. If neither, find the angle between them.
18. Find the equation of the plane consisting of all points that are equidistant from the points $\left(-4,2,1\right)$ and $\left(2,-4,3\right)$.
19. Determine whether each statement is true or false in ${ℝ}^3$.

    1. Two lines parallel to a third line are parallel.
    2. Two lines perpendicular to a third line are parallel.
    3. Two planes parallel to the third plane are parallel.
    4. Two planes perpendicular to the third plane are parallel.
    5. Two lines parallel to the same plane are parallel.
    6. Two lines perpendicular to the same plane are parallel.
    7. Two planes parallel to the same line are parallel.
    8. Two planes perpendicular to the same line are parallel.
    9. Two planes either intersect or are parallel.
    10. Two lines either intersect or are parallel.
    11. A plane and a line either intersect or are parallel.
    True, false, true, false, false, true, false, true, true, false, true.
20. Find parametric equations and symmetric equations for the line of intersection of the planes $x+y+z=1$ and $x+z=0$.

21. Which of the following planes are parallel? Which are identical?

    $P_1$: $4x-2y+6z=3$

    $P_2$: $4x-2y-2z=6$

    $P_3$: $-6x+3y-9z=5$

    $P_4$: $z=2x-y-3$

    $P_1$ and $P_3$ are parallel, $P_2$ and $P_4$ are identical.

22. Which of the following lines are parallel? Which are identical?

    $L_1$: $x=1+t$, $y=t$, $z=2-5t$

    $L_2$: $x+1=y-2=1-z$

    $L_3$: $x=1+t$, $y=4+t$, $z=1-t$

    $L_4$: $\mathbf{r}=⟨2,1,-3⟩+t⟨2,2,-10⟩$

23. Find the distance from the point $\left(2,8,5\right)$ to the plane $x-2y-2z=1$.
    ${\displaystyle \frac{25}{3}}$
24. Find the distance between (parallel) planes $x+2y-3z=1$ and $3x+6y-9z=4$.
    ${\displaystyle \frac{1}{3\sqrt{14}}}$
25. Find equations of the planes that are parallel to the plane $x+2y-2z=1$ and $2$ units away from it.
26. Show that the lines with symmetric equations $x=y=z$ and ${\displaystyle x+1=\frac{y}{2}=\frac{z}{3}}$ are skew, and find the distance between these lines.
    ${\displaystyle \frac{1}{\sqrt{6}}}$

27. Find the angle between the lines

    ${\mathbf{L}}_1<!--FUNCTION\; APPLICATION-->\left(t\right)=⟨1+t,3+2t,1+t⟩$ and ${\mathbf{L}}_2<!--FUNCTION\; APPLICATION-->\left(t\right)=⟨1+t,3+t,1+2t⟩$.

    ${\displaystyle {\cos }^{-1}<!--FUNCTION\; APPLICATION-->\left(\frac{5}{6}\right)}$
28. Compute all unit vectors orthogonal to the plane $5=2x+4y-z$.
    ${\displaystyle \pm \frac{1}{\sqrt{21}}⟨2,4,-1⟩}$
29. Find an equation for the plane containing the point $\left(1,-1,1\right)$ and the line $x=2y=3z$.
    $-5x+4y+9z=0$.