Show that the cancellation law does not hold for the dot product

by RoRi

April 26, 2016

Prove or disprove the following statement about vectors in $\mathbb{R}^n$ :

If $A \cdot B = A \cdot C$ and $A \neq O$ , then $B = C$ .

This statement is false. Let $A = (1,1,1)$ , $B = (1,-1,0)$ and $C = (0,-1,1)$ . Then we have

$A \cdot B = (1)(1)+(1)(-1) + (1)(0) = 0, \qquad A \cdot C = (1)(0) + (1)(-1) + (1)(1) = 0$

but, $B \neq C$ .

Stumbling Robot