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\begin{pmatrix}3\\ 1\\ 2\\0\end{pmatrix}\begin{pmatrix}4 &-1 &0 &1 \end{pmatrix}

 

3\cdot 4+1\cdot -1+2\cdot 0+0\cdot 1=12-1+0+0=11

 


\begin{pmatrix}x\\ y\\ z\end{pmatrix}\begin{pmatrix}-x &3 &z \end{pmatrix}

 

x\cdot -x+y\cdot 3+z\cdot z=-x^{2}+3y+z^{2}

 


\begin{pmatrix}\sqrt{2}\\ \frac{1}{2}\\ 5\end{pmatrix}\begin{pmatrix}\sqrt{2} &6 &m \end{pmatrix}

 

\sqrt{2}\cdot \sqrt{2}+\frac{1}{2}\cdot 6+5\cdot m=2+3+5m=5m+5