Show vector exists in the plane

Question

Let $\{a, b\}$ a linearly independent set in $\mathbb{R}^3$. If $c \in \mathbb{R}^3$ such that $(a\times b)\cdot c= 0$, then $c \in \text{span } \{a, b\}$

I know this is true but I cant prove it. How do I show that $c$ is a vector in the plane $P = \text{span } \{a, b\}$?

Answer 1

Hint: Clearly $a,b,a\times b$ is a basis of your space. Now write $c$ as a linear combination of the above basis and take the inner product with $a\times b$.

Rmk: The following question is quite related Prove that $\{v_1, v_2, n\}$ is a basis