Equations of orthogonal planes containing a given line

Question

Having difficulty with this problem:

Find equations for two orthogonal planes, both of which contain the line $\mathbf{v}=(1,0,3) + t(-1,2,1)$, one of which passes through the origin.

Answer 1

Let the plane passes through the origin be $\Pi_{1}: \: ax+by+cz=0$

Now $$(a,b,c)\cdot (-1,2,1)=0$$

$$-a+2b+c=0$$

Also $\Pi_{1}$ contains $(1,0,3)$,

$$a+3c=0$$

Therefore $$a:b:c=3:2:-1$$

Now another plane with normal parallel to $$(3,2,-1)\times(-1,2,1)=(4,-2,8)$$

Let $\Pi_{2}: \: 2x-y+4z=d$ and $\Pi_{2}$ contains $(1,0,3)$

$$2(1)-0+4(3)=d$$

\begin{array}{rrcl} \Pi_{1}: && 3x+2y-z &=& 0 \\ \Pi_{2}: && 2x-y+4z &=& 14 \end{array}