Let \(u = \begin{bmatrix} 1 \\-2 \\1\end{bmatrix}\) and let \(v = \begin{bmatrix} 3 \\ 1 \\ -1\end{bmatrix}\).
What is \(u\cdot v\)?
The answer is 0.
\begin{eqnarray} u\cdot v & = & u_1v_1 + u_2v_2 + u_3v_3 \\ & = & (1)(3)+(-2)(1)+(1)(-1) \\ & = & 3-2-1 \\ & = & 0. \end{eqnarray}