(a) If \(\mathbf{v}=2\mathbf{i}-3\mathbf{j}\) and \(\mathbf{w}= \mathbf{i}+\mathbf{j}\), then \[ \begin{equation*} \begin{array}{rcl@{\quad}rcl} \mathbf{v}\,{\cdot}\, \mathbf{w} &=& (2)(1)+(-3)(1)=2-3=-1 & \mathbf{w\,{\cdot}\, v} &=& (1) (2) +(1) (-3) =2-3=-1 \\ \mathbf{v\,{\cdot}\, v} &=& 2^{2}+ (-3) ^{2}=4+9=13 & \mathbf{w\,{\cdot}\, w} &=& 1^{2}+1^{2}=2 \end{array} \end{equation*} \]
(b) If \(\mathbf{v}=2\mathbf{i}-\mathbf{j}+\mathbf{k}\) and \( \mathbf{w}=4\mathbf{i}+2\mathbf{j}-\mathbf{k}\), then \[ \begin{array}{rcl@{\quad}rcl} \mathbf{v}\,{\cdot}\, \mathbf{w}&=&(2)(4)+(-1)(2)+(1)(-1) & \mathbf{w\,{\cdot}\, v}&=&(4) (2) +(2) (-1) +(-1) (1) \\ &=&8-2-1=5 & &=&8-2-1=5\\ \mathbf{v}\,{\cdot}\, \mathbf{v}&=&4+1+1=6 & \mathbf{w}\,{\cdot}\, \mathbf{w}&=&16+4+1=21 \end{array} \]