Answer
\[\begin{align}
& \mathbf{a}.\left\langle 12,-7,2 \right\rangle \\
& \mathbf{b}.\left\langle 16,-13,-1 \right\rangle \\
& \mathbf{c}.5 \\
\end{align}\]
Work Step by Step
\[\begin{align}
& \mathbf{u}=\left\langle 4,-3,0 \right\rangle ,\text{ }\mathbf{v}=\left\langle 0,1,1 \right\rangle \\
& \mathbf{a}.\text{ 3}\mathbf{u}+\text{2}\mathbf{v} \\
& \text{3}\mathbf{u}+\text{2}\mathbf{v}=3\left\langle 4,-3,0 \right\rangle +2\left\langle 0,1,1 \right\rangle \\
& \text{3}\mathbf{u}+\text{2}\mathbf{v}=\left\langle 12,-9,0 \right\rangle +\left\langle 0,2,2 \right\rangle \\
& \text{3}\mathbf{u}+\text{2}\mathbf{v}=\left\langle 12+0,-9+2,0+2 \right\rangle \\
& \text{3}\mathbf{u}+\text{2}\mathbf{v}=\left\langle 12,-7,2 \right\rangle \\
& \\
& \mathbf{b}.\text{ 4}\mathbf{u}-\mathbf{v} \\
& \text{4}\mathbf{u}-\mathbf{v}=4\left\langle 4,-3,0 \right\rangle -\left\langle 0,1,1 \right\rangle \\
& \text{4}\mathbf{u}-\mathbf{v}=\left\langle 16,-12,0 \right\rangle -\left\langle 0,1,1 \right\rangle \\
& \text{4}\mathbf{u}-\mathbf{v}=\left\langle 16-0,-12-1,0-1 \right\rangle \\
& \text{4}\mathbf{u}-\mathbf{v}=\left\langle 16,-13,-1 \right\rangle \\
& \\
& \mathbf{c}.\text{ }\left| \mathbf{u}+3\mathbf{v} \right| \\
& \left| \mathbf{u}+3\mathbf{v} \right|=\left| \left\langle 4,-3,0 \right\rangle +3\left\langle 0,1,1 \right\rangle \right| \\
& \left| \mathbf{u}+3\mathbf{v} \right|=\left| \left\langle 4,-3,0 \right\rangle +\left\langle 0,3,3 \right\rangle \right| \\
& \left| \mathbf{u}+3\mathbf{v} \right|=\left| \left\langle 4,0,3 \right\rangle \right| \\
& \left| \mathbf{u}+3\mathbf{v} \right|=\sqrt{{{\left( 4 \right)}^{2}}+{{\left( 0 \right)}^{2}}+{{\left( 3 \right)}^{2}}} \\
& \left| \mathbf{u}+3\mathbf{v} \right|=\sqrt{16+0+9} \\
& \left| \mathbf{u}+3\mathbf{v} \right|=5 \\
\end{align}\]
