Chapter 11 - Vectors and Vector-Valued Functions - 11.2 Vectors in Three Dimensions - 11.2 Exercises - Page 779: 41

\[\begin{align} & \mathbf{a}.\left\langle -4,5,-4 \right\rangle \\ & \mathbf{b}.\left\langle -9,3,-9 \right\rangle \\ & \mathbf{c}.3\sqrt{2} \\ \end{align}\]

\[\begin{align} & \mathbf{u}=\left\langle -2,1,-2 \right\rangle ,\text{ }\mathbf{v}=\left\langle 1,1,1 \right\rangle \\ & \mathbf{a}.\text{ 3}\mathbf{u}+\text{2}\mathbf{v} \\ & \text{3}\mathbf{u}+\text{2}\mathbf{v}=3\left\langle -2,1,-2 \right\rangle +2\left\langle 1,1,1 \right\rangle \\ & \text{3}\mathbf{u}+\text{2}\mathbf{v}=\left\langle -6,3,-6 \right\rangle +\left\langle 2,2,2 \right\rangle \\ & \text{3}\mathbf{u}+\text{2}\mathbf{v}=\left\langle -4,5,-4 \right\rangle \\ & \\ & \mathbf{b}.\text{ 4}\mathbf{u}-\mathbf{v} \\ & \text{4}\mathbf{u}-\mathbf{v}=4\left\langle -2,1,-2 \right\rangle -\left\langle 1,1,1 \right\rangle \\ & \text{4}\mathbf{u}-\mathbf{v}=\left\langle -8,4,-8 \right\rangle -\left\langle 1,1,1 \right\rangle \\ & \text{4}\mathbf{u}-\mathbf{v}=\left\langle -8-1,4-1,-8-1 \right\rangle \\ & \text{4}\mathbf{u}-\mathbf{v}=\left\langle -9,3,-9 \right\rangle \\ & \\ & \mathbf{c}.\text{ }\left| \mathbf{u}+3\mathbf{v} \right| \\ & \left| \mathbf{u}+3\mathbf{v} \right|=\left| \left\langle -2,1,-2 \right\rangle +3\left\langle 1,1,1 \right\rangle \right| \\ & \left| \mathbf{u}+3\mathbf{v} \right|=\left| \left\langle -2,1,-2 \right\rangle +\left\langle 3,3,3 \right\rangle \right| \\ & \left| \mathbf{u}+3\mathbf{v} \right|=\left| \left\langle 1,4,1 \right\rangle \right| \\ & \left| \mathbf{u}+3\mathbf{v} \right|=\sqrt{{{\left( 1 \right)}^{2}}+{{\left( 4 \right)}^{2}}+{{\left( 1 \right)}^{2}}} \\ & \left| \mathbf{u}+3\mathbf{v} \right|=\sqrt{1+16+1} \\ & \left| \mathbf{u}+3\mathbf{v} \right|=\sqrt{18} \\ & \left| \mathbf{u}+3\mathbf{v} \right|=3\sqrt{2} \\ \end{align}\]