12.5 Videos Guide

12.5a

• Equations of a line in space parallel to the vector $\mathbf{v}=\left\langle a,\ b,\ c\right\rangle$ and containing the point $(x_{0},y_{0},\ z_{0})$

  • Vector equation: $\mathbf{r}(t)=\left\langle x_{0},y_{0},x_{0}\right\rangle +t\left\langle a,\ b,\ c\right\rangle$

  • Parametric equations: $x=x_{0}+at$, $y=y_{0}+bt$, $z=z_{0}+ct$

  • Symmetric equations: $\frac{x-x_{0}}{a}=\frac{y-y_{0}}{b}=\frac{z-z_{0}}{c}$

Exercise:

• Find a vector equation and parametric equations for the line through the point $(6,\ -5,\ 2)$ and parallel to the vector $\left\langle 1,\ 3,\ -\frac{2}{3}\right\rangle$.

12.5b

• Vector equation of a line segment from $P(x_{0},y_{0},z_{0})$ to $Q(x_{1},y_{1},z_{1})$:
  Let $\mathbf{r}_{0}(t)=\left\langle x_{0},y_{0},z_{0}\right\rangle$ and $\mathbf{r}_{1}(t)=\left\langle x_{1},y_{1},z_{1}\right\rangle$

  • $\mathbf{r}(t)=(1-t)\mathbf{r}_{0}+t\mathbf{r}_{1}$, $0\leq t\leq1$

12.5c

• Equation of a plane with normal vector $\mathbf{n}=\left\langle a,\ b,\ c\right\rangle$ and containing the point $(x_{0},y_{0},z_{0})$

  • $a\left(x-x_{0}\right)+b\left(y-y_{0}\right)+c\left(z-z_{0}\right)=0$

Exercises:

• Visually represent the portion of the plane $3x+4y+6z=12$ that is in the first octant.

12.5d

• Find an equation of the plane through the points $(3,\ 0,\ -1)$, $(-2,\ -2,\ 3)$, and $(7,\ 1,\ -4)$.

12.5e

• Find an equation of the plane that passes through the point $(6,\ -1,\ 3)$ and contains the line with symmetric equations $\frac{x}{3}=y+4=\frac{z}{2}$.

• Where does the line through $(-3,\ 1,\ 0)$ and $(-1,\ 5,\ 6)$ intersect the plane
  $2x+y-z=-2$?

12.5f

• (a) Find parametric equations for the line of intersection of the planes and (b) find the angle between the planes.
  $x+y+z=1$, $x+2y+2z=1$

12.5g

• Distance from a point $(x_{1},y_{1},z_{1})$ to a plane with equation $ax+by+cz+d=0$

  • $D=\frac{\left|ax_{1}+by_{1}+cz_{1}+d\right|}{\sqrt{a^{2}+b^{2}+c^{2}}}$

Exercises:

• Find the distance from the point $(-6,\ 3,\ 5)$ to the plane $x-2y-4z=8$.

12.5h

• Find the distance between the skew lines with the given parametric equations.
  $x=1+t$ and $x=1+2s$
  $y=1+6t$ $y=5+15sz=2t$ $z=-2+6s$