# Thread: equation of a line through a point, parallel to a plane, and perpendicular to a line

1. ## equation of a line through a point, parallel to a plane, and perpendicular to a line

Q: Find parametric equations for the line through the point (0, 1, 2) that is parallel to the plane $\displaystyle x+y+z=2$ and perpendicular to the line $\displaystyle x=1+t, y=1-t, z=2t$

I'm having trouble coming up with a sensible approach to this problem (my understanding of point/line/plane interaction is a bit shaky as solutions to similar problem have been found by applying methods used in the book examples)

I know an equation of a line is given by $\displaystyle \vec{r}=\vec{r_{0}}+t\vec{v}$ where $\displaystyle \vec{r_{0}}$ is position vector of a point on the line and $\displaystyle \vec{v}$ is a vector parallel with the line. The vector,$\displaystyle \vec{n}=[1, 1, 1]$, is the normal vector for the parallel plane, and $\displaystyle \vec{v}=[1, -1, 2]$ is the vector in the direction of the perpendicular line. Is this all I need to solve? If so, where do I go from here?

2. ## Re: equation of a line through a point, parallel to a plane, and perpendicular to a l Originally Posted by sgcb Q: Find parametric equations for the line through the point (0, 1, 2) that is parallel to the plane $\displaystyle x+y+z=2$ and perpendicular to the line $\displaystyle x=1+t, y=1-t, z=2t$
The direction vector you need is $\displaystyle <1,1,1>\times<1,-1,2>$.

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