What is the equation of a 3D line which represents the intersection between two 3D planes? The intersection defined by the two planes  $v \bullet \begin{pmatrix} 8 \\ 1 \\ -12 \end{pmatrix} = 35$
and
$v \bullet \begin{pmatrix} 6 \\ 7 \\ -9 \end{pmatrix} = 70$
is a line. What is the equation of this line?
This is what I have so far: I set $v = \begin{pmatrix} a \\ b \\ c \end{pmatrix}$. I was able to simplify both LHS of the two given equations to: 
$8a+b-12c = 35$
$6a+7b-9c = 70$
I could find the values of $a, b, c$, but I don't know if it will be helpful. How can I find the equation of the line of intersection?
 A: There are two basic methods: either


*

*You can obtain the direction of the line by doing the cross-product of the normals $$\left(\begin {matrix}8\\1\\-12\end {matrix}\right)\times\left(\begin {matrix}6\\7\\-9\end {matrix}\right)=\left(\begin {matrix}75\\0\\50\end {matrix}\right) \text{which is parallel to}\left(\begin {matrix}3\\0\\2\end {matrix}\right)$$


And then somehow obtain a point which satisifies both plane equations, perhaps by assigning a value to one of the variables and solving for the other two.
Or, a more systematic way is:


*Start with your plane equations (I am using $x$, $y$ and $z$) and eliminate any one of the variables,
So, for example, if you eliminate $y$ you get $$2x-3z=7$$
Now separate the letters onto opposite sides of the equation and intoduce $=\lambda$. So now you have $$2x=7+3z=\lambda$$


Now get each of $x$, $y$ and $z$ in terms of $\lambda$ and you end up with the equation of the line in one of many possible vector forms; in this case $$\underline{r}=\left(\begin {matrix}0\\7\\\frac{-7}{3}\end {matrix}\right)+\lambda \left(\begin {matrix}\frac 12\\0\\\frac 13\end {matrix}\right)$$
Of course you can write the direction vector as $$\left(\begin {matrix}3\\0\\2\end {matrix}\right)$$ as before.
A: With two equations and three unknowns your solution space will be a one-dimensional set (a line), exactly what you are looking for. You will need to write $(a,b,c)$ in terms of one of the parameters. For instance $a=f_{1}(c),b=f_{2}(c),c=c$.
It will not always be the case that a solution space will be one-dimensional, it could be the case that the column space is one-dimensional and it is possible that no solution would exist.
