Let A be a vector whose components are the real numbers $a_k\,k\=1\,\dots \,n$, and assume $\mathbf{A}·\mathbf{A}\=0$.

Now $\mathbf{A}·\mathbf{A}\=\sum _{k\=1}^n{a}_k^2$, so the condition that $\mathbf{A}·\mathbf{A}\=0$ means a sum of positive real numbers adds to zero, something that can only happen if each of the numbers is itself zero. Hence, all the components of A are zero, and that makes A the zero vector.

Now assume that A is the zero vector so that all its components are zero. Since $\mathbf{A}·\mathbf{A}\=\sum _{k\=1}^n{a}_k^2$ and all the $a_k$ are zero, the sum must be zero, and therefore $\mathbf{A}·\mathbf{A}\=0$.