Vector equation of a line

Question

The line $l$ has an equation $r=\begin{pmatrix} 1\\ 2\\ -1 \end{pmatrix}+\lambda \begin{pmatrix} 2\\ 1\\3 \end{pmatrix}$ and equation of plane $p= r.\begin{pmatrix} 2\\ -1\\ -1 \end{pmatrix}$

i) show that line $l$ is parallel to plane $p$

I managed to solve this question by showing scalar product of direction vector of line and normal of plane is $0$

ii)A line $m$ lies in the plane $p$ and is perpendicular to $l$. The line $m$ passes through the point $(5,3,1)$ .Find equation of line $m$.

I know that the equation of the line will be in the form $\begin{pmatrix} 5\\ 3\\ 1 \end{pmatrix}+direction$

How to find the direction , Please assist , also line $m$ lies on plane and $l$ is parallel to plane , how can $m$ be perpendicular to $l$, Ps assist.

thank you, Arif

Answer 1

HINT

Let $d$ be the direction of line $m$. Because line $m$ is in the plane, then $d$ is perpendicular on the normal of the plane.

Because line $m$ is perpendicular on $l$ then $d$ is perpendicular on the direction of $l$.

Take into account that perpendicular lines in space may not intersect.

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