Lines and planes in space

Write the Cartesian equation of the plane passing through the points $A=(1,2,0)$, $B=(3,0,1)$, $C=(0,1,1)$.

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Write the equation of the plane passing through $P=(1,-1,2)$ and parallel to the lines

$$r:x=1+t,y=2t,z=3-t,s:x=2+u,y=1,z=u.$$

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Write the parametric and Cartesian equations of the line passing through $A=(2,1,-1)$ and $B=(4,0,1)$.

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Write the parametric equations of the line

$$r:\{\begin{array}{l}x+2y-z=3\\ x-y+z=0\end{array}$$

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As $k\in \mathbb{R}$ varies, determine the relative position of the line $r:x=1+t$, $y=1-t$, $z=2+t$ and the plane ${\pi }_k:x+2y+kz=4$; for $k=0$ find the common point.

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Determine the relative position of the lines

$$r:x=1+2t,y=-1+t,z=t,s:\{\begin{array}{l}x-z=1\\ y+z=-1\end{array}$$

and, if they are intersecting, find their common point.

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Compute the angle between the planes ${\pi }_1:2x+y+2z=1$ and ${\pi }_2:y+z=4$.

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Compute the angle between the line

$$r:\{\begin{array}{l}x-y+1=0\\ z=2\end{array}$$

and the plane $\pi :y-z=3$.

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Compute the distance from the point $P=(2,-1,3)$ to the plane $\pi :x-2y+2z+1=0$.

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Compute the distance from the point $P=(4,1,2)$ to the line $r:x=2t$, $y=1+2t$, $z=1+t$ and find the orthogonal projection of $P$ onto $r$.

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Find the orthogonal projection $H$ of the point $P=(3,-1,2)$ onto the plane $\pi :2x-y+z=3$ and the symmetric $P^{\prime }$ of $P$ with respect to $\pi$.

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Find the plane that contains the line

$$r:\{\begin{array}{l}x-y+z=1\\ 2x+y-z=2\end{array}$$

and is parallel to the line $s:x=t$, $y=1+2t$, $z=3-t$.

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Given the skew lines $r:x=1+t$, $y=2+t$, $z=1$ and $s:x=1+u$, $y=-1$, $z=-2+u$, find the distance between the lines and the equations of the line of minimum distance.

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