Find v3 so that v1 v2 v3 is a basis of R^3

Question

Let $v_1=(2,3,4),v_2=(1,0,2)$. Find $v_3$ so that $\{v_1, v_2, v_3\}$ is a basis of $\mathbb{R}^3$?

I don't really know how to solve and it's a common question, I would really appreciate the help. Thank you.

Answer 1

Note that $v_1$ and $v_2$ are linearly independent and in $\mathbb{R}^3$. You need a third vector which is linearly independent of the first two. The easiest way in $\mathbb{R}^3$ is to take $$ v_1 \times v_2 $$ which is perpendicular to both $v_1$ and $v_2$ and hence linearly independent of both.