# Vectors: Two lines that intersect, one vector that is perpendicular to both

So I have this maths question that I can't seem to wrap my head around. I have two lines:

Line 1 $$= r = 3i+2j+7k + X(i-j+3)$$

Line 2 $$= r = 6i+5j+2k + Y(2i+j-k)$$

I have found the point of intersection when $$X=-1$$ and $$Y=-2$$, and thus the point of intersection to be $$2i+3j+4k$$

Now I have been told that a vector $$(i+aj+bk)$$ is perpendicular to both, and to find $$a$$ and $$b$$. I understand how this is, but can't seem to find the answer. I can't use dot product with two unknowns, so I would really appreciate some help.

P.S I'm not sure how you're properly supposed to structure these questions with the fancy format so if anyone could link me to how I am supposed to lay questions out please do.

• Hello @Harvey Stanfield, here is the link that explains how to use MathJax. – Ernie060 Oct 22 '18 at 21:34

• Your method works. My idea was the desired vector should be some scalar multiple of the cross product of the direction vectors. Then adjust to match first component $i$. [should give same answer, but either way OK.] (+1 on your answer) – coffeemath Oct 22 '18 at 22:17