Tag: dot product

Prove that the only vectors whose dot product with all vectors is zero is the zero vector

by RoRi

April 26, 2016

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

If $A \cdot B = O$ for every vector $B \in \mathbb{R}^n$ , then $A = O$ .

Proof. Let $A,B \in \mathbb{R}^n$ . If $A \cdot B = O$ for every $B$ we must, in particular, have $A \cdot A = O$ . But $A \cdot A = O$ if and only if $A = O$ . Hence, we must have $A = O. \qquad \blacksquare$