For the vectors \( \mathbf{u}=\langle 3,2,4 \sqrt{3}\rangle \) and \( \mathbf{v}=\langle 1,0,4 \sqrt{3}\rangle \), evaluate the following expressions. \( \begin{array}{lll}\text { a. } 2 \mathbf{u}+3 \mathbf{v} & \text { b. } 4 \mathbf{u}-\mathbf{v} & \text { c. }|\mathbf{u}+2 \mathbf{v}|\end{array} \) \( \begin{array}{l}\text { a. } 2 \mathbf{u}+3 \mathbf{v}= \\ \text { (Type exact answers, using radicals as needed.) }\end{array} \)

Let's evaluate each expression step by step. **a.** To find \( 2 \mathbf{u} + 3 \mathbf{v} \): First, calculate \( 2 \mathbf{u} \) and \( 3 \mathbf{v} \): \[ 2 \mathbf{u} = 2 \langle 3, 2, 4\sqrt{3} \rangle = \langle 6, 4, 8\sqrt{3} \rangle \] \[ 3 \mathbf{v} = 3 \langle 1, 0, 4\sqrt{3} \rangle = \langle 3, 0, 12\sqrt{3} \rangle \] Now, add the two results together: \[ 2 \mathbf{u} + 3 \mathbf{v} = \langle 6, 4, 8\sqrt{3} \rangle + \langle 3, 0, 12\sqrt{3} \rangle = \langle 6 + 3, 4 + 0, 8\sqrt{3} + 12\sqrt{3} \rangle \] This simplifies to: \[ = \langle 9, 4, 20\sqrt{3} \rangle \] So, \[ \text{a. } 2 \mathbf{u} + 3 \mathbf{v} = \langle 9, 4, 20\sqrt{3} \rangle \] **b.** To find \( 4 \mathbf{u} - \mathbf{v} \): First, calculate \( 4 \mathbf{u} \) and \( \mathbf{v} \): \[ 4 \mathbf{u} = 4 \langle 3, 2, 4\sqrt{3} \rangle = \langle 12, 8, 16\sqrt{3} \rangle \] Now, subtract \( \mathbf{v} \): \[ 4 \mathbf{u} - \mathbf{v} = \langle 12, 8, 16\sqrt{3} \rangle - \langle 1, 0, 4\sqrt{3} \rangle = \langle 12 - 1, 8 - 0, 16\sqrt{3} - 4\sqrt{3} \rangle \] This simplifies to: \[ = \langle 11, 8, 12\sqrt{3} \rangle \] So, \[ \text{b. } 4 \mathbf{u} - \mathbf{v} = \langle 11, 8, 12\sqrt{3} \rangle \] **c.** Now, let's calculate \( |\mathbf{u} + 2 \mathbf{v}| \): First, calculate \( 2 \mathbf{v} \): \[ 2 \mathbf{v} = 2 \langle 1, 0, 4\sqrt{3} \rangle = \langle 2, 0, 8\sqrt{3} \rangle \] Now, add \( \mathbf{u} \) to this: \[ \mathbf{u} + 2 \mathbf{v} = \langle 3, 2, 4\sqrt{3} \rangle + \langle 2, 0, 8\sqrt{3} \rangle = \langle 3 + 2, 2 + 0, 4\sqrt{3} + 8\sqrt{3} \rangle \] This simplifies to: \[ = \langle 5, 2, 12\sqrt{3} \rangle \] Now, find the magnitude: \[ |\mathbf{u} + 2 \mathbf{v}| = \sqrt{5^2 + 2^2 + (12\sqrt{3})^2} = \sqrt{25 + 4 + 432} = \sqrt{461} \] So, \[ \text{c. } |\mathbf{u} + 2 \mathbf{v}| = \sqrt{461} \] Now putting them all together, we have: \[ \text{a. } \langle 9, 4, 20\sqrt{3} \rangle \\ \text{b. } \langle 11, 8, 12\sqrt{3} \rangle \\