Obtain the equation of the plane containing the three points $\left(1\,2\,3\right)$, $\left(-1\,3\,1\right)$, $\left(2\,1\,-1\right)$.

$\left[1\,2\,3\right]\,\left[-1\,3\,1\right]\,\left[2\,1\,-1\right]$$\stackrel{\text{make plane}}{\to }$$\mathrm{Student}:-\mathrm{MultivariateCalculus}:-\mathrm{Plane}\left(\left[\begin{array}{r}-6\\ -10\\ 1\end{array}\right]\,\left[1\,2\,3\right]\,\mathrm{variables}\=\left[x\,y\,z\right]\,\mathrm{id}\=1\right)$$\stackrel{\text{representation}}{\to }$$-6x-10y+z=\mathrm{-23}$

Show that $x\=1\+2 t\,y\=2-3 t\,z\=3\+5 t$ and $x\=3-s\,y\=5\+3\,z\=7\+6 s$ define skew lines, and find the distance between them.

Create Line Objects for each line

$\left[x\=1\+2 t\,y\=2-3 t\,z\=3\+5 t\right]$$\stackrel{\text{make line}}{\to }$$\mathrm{Student}:-\mathrm{MultivariateCalculus}:-\mathrm{Line}\left(\left[1\,2\,3\right]\,\left[\begin{array}{r}2\\ -3\\ 5\end{array}\right]\,\mathrm{variables}\=\left[x\,y\,z\right]\,\mathrm{parameter}\=t\,\mathrm{id}\=1\right)$$\stackrel{\text{assign to a name}}{\to }$$\mathrm{L1}$

$\left[x\=3-s\,y\=5\+3\,z\=7\+6 s\right]$$\stackrel{\text{make line}}{\to }$$\mathrm{Student}:-\mathrm{MultivariateCalculus}:-\mathrm{Line}\left(\left[3\,8\,7\right]\,\left[\begin{array}{r}-1\\ 0\\ 6\end{array}\right]\,\mathrm{variables}\=\left[x\,y\,z\right]\,\mathrm{parameter}\=s\,\mathrm{id}\=2\right)$$\stackrel{\text{assign to a name}}{\to }$$\mathrm{L2}$

Verify the lines are skew

Obtain the distance between the lines

$\mathrm{L1}\,\mathrm{L2}$$\stackrel{\text{distance}}{\to }$$\frac{75\sqrt{622}}{311}$$\stackrel{\text{at 10 digits}}{\to }$$6.014452050$

Obtain N, the common normal