For $\mathbf{u}=\langle 3,-1\rangle, \mathbf{v}=\langle 3,1\rangle$, and $\mathbf{w}=\langle 1,3\rangle$, evaluate the expression $(4 u) \cdot v$ $(4 u) \cdot v=\square($ Simplify your answer.)

Step 1 ：Given vectors are \(\mathbf{u} = \langle 3,-1 \rangle\), \(\mathbf{v} = \langle 3,1 \rangle\), and \(\mathbf{w} = \langle 1,3 \rangle\)

Step 2 ：We need to evaluate the expression \((4\mathbf{u}) \cdot \mathbf{v}\)

Step 3 ：First, multiply the vector \(\mathbf{u}\) by 4 to get \(4\mathbf{u} = \langle 12,-4 \rangle\)

Step 4 ：Then, calculate the dot product of \(4\mathbf{u}\) and \(\mathbf{v}\) by multiplying the corresponding entries of the two vectors and then adding those products

Step 5 ：So, \((4\mathbf{u}) \cdot \mathbf{v} = (12*3) + (-4*1) = 36 - 4 = 32\)

Step 6 ：Final Answer: \((4\mathbf{u}) \cdot \mathbf{v} = \boxed{32}\)