Find a nonzero vector satisfying a requested property

by RoRi

April 26, 2016

Let

$A = (2,1,-1), \qquad B = (1,-1,2).$

Find some vector $C \neq O$ in $\mathbb{R}^3$ such that

$A \cdot C = B \cdot C = 0.$

Let $C = (c_1, c_2, c_3)$ . Then we have

$\begin{align*} A \cdot C &= 2c_1 + c_2 - c_3 = 0\\ B \cdot C &= c_1 - c_2 + 2c_3 = 0. \end{align*}$

From these equations we have

$c_1 = \frac{1}{2} (c_3 - c_2) \quad \implies \quad \frac{5}{2} c_3 - \frac{3}{2}c_2 = 0 \quad \implies \quad c_2 = \frac{5}{3} c_3.$

Letting $c_3 = 3$ we then have $c_2 = 5$ and $c_1 = -1$ . Thus, $C = (-1,5,3)$ is a solution.

Stumbling Robot