Let $W_1$ and $W_2$ be subspaces of $\mathbb{R}^3$ under addition and scalar multiplication such that:
$$W_1=\{(a_1,a_2,a_3)\in \mathbb{R}^3:2a_1-7a_2+a_3=0\}$$
$$W_2=\{(a_1,a_2,a_3)\in \mathbb{R}^3:a_1-4a_2-a_3=0\}$$
Describe $W_1\cap W_2$
My thoughts:
Clearly $W_1$ and $W_2$ are planes with normal vectors $(2,-7,1)$ and $(1,-4,-1)$, respectively.
It must be that their intersection is a line, but I don't know how to show this or express the equation of this line.
I tried setting their planar equations equal:
$$2a_1-7a_2+a_3=a_1-4a_2-a_3$$ $$ \Rightarrow a_1-3a_2+2a_3=0$$
But is this not just another plane, which obviously can't be correct? I think there is something simple that I am overlooking.
For reference, this question is from "Linear Algebra Second Edition" by Friedberg, Insel, Spence, Chapter 1 Section 3 Exercise 9
Any hints or guidance would be greatly appreciated.