Show a, b and c are coplanar

Question

A plane contains the non-zero, non-parallel vectors a and b, and has normal vector n. If c is any vector which is perpendicular to n, show that a, b and c are coplanar.

How would I go about showing this without having any position vectors?

Answer 1

Recall that

$$\vec a \times \vec b = k\vec n$$

let $\vec c$ such that $\vec c \cdot \vec n =0$ than

$$\vec c \cdot \vec n=\vec c \cdot (\vec a \times \vec b)=0$$

and since the triple product is equal to zero, we have that $\vec a$, $\vec b$ and $\vec c$ are coplanars.