Find scalars such that given vector equations are satisfied

by RoRi

April 26, 2016

Let

$A = (1,-2,3), \qquad B = (1,2,-2).$

Find scalars $x,y \in \mathbb{R}$ such that

$C =xA + yB$

with $C$ nonzero and $C \cdot B = 0$ .

First, we have

$C = xA +yB \quad \implies \quad C = (x+3y, -2x+y, 3x+2y),$

and

$B \cdot C = 0 \quad \implies \quad 3(x+3y) + (-2x+y) + 2(3x+2y) = 0 \quad \implies \quad x + 2y = 0.$

Let $x = -2$ and $y = 1$ is a then a solution.

Stumbling Robot