Find the parametric equation of the line through the point p = (0,1,2) that is parallel to the plane x+y+z=2 and perpendicular to the line x=1+t , y=1-t , z=2t.
Hello, tttcomrader!
We need only the direction vector of our line, $\displaystyle \vec{v}$Find the parametric equation of the line through the point $\displaystyle P(0,1,2)$
that is parallel to the plane $\displaystyle x+y+z\:=\:2$ and perpendicular to the line: .$\displaystyle \begin{array}{ccc}x&=&1+t \\ y&=&1-t \\ z&=&2t\end{array}$
To be parallel to the plane, $\displaystyle \vec{v}$ is perpendicular to the normal vector of the plane: $\displaystyle \vec{n} \:=\:\langle 1,1,1\rangle$
To be perpendicular to the line, $\displaystyle \vec{v}$ is perpendicular to the vector $\displaystyle \vec{u} \:=\:\langle 1,\,-1,\,2\rangle$
Hence: .$\displaystyle \vec{v} \;=\;\vec{n} \times \vec{u}$
. . . Go for it!