For $\mathbf{u}=\langle 3,-2\rangle$ and $\mathbf{v}=\langle-1,2\rangle$, find $\mathbf{u} \cdot \mathbf{v}$. \[ \mathbf{u} \cdot \mathbf{v}= \]

Step 1 ：Given vectors \(\mathbf{u}=\langle 3,-2\rangle\) and \(\mathbf{v}=\langle-1,2\rangle\), we are asked to find \(\mathbf{u} \cdot \mathbf{v}\).

Step 2 ：The dot product of two vectors is calculated by multiplying the corresponding components of the vectors and then adding those products together.

Step 3 ：In this case, we need to multiply the first component of vector u by the first component of vector v, and then add that to the product of the second component of vector u and the second component of vector v.

Step 4 ：So, \(\mathbf{u} \cdot \mathbf{v} = (3 \times -1) + (-2 \times 2) = -3 - 4 = -7\).

Step 5 ：Final Answer: \(\mathbf{u} \cdot \mathbf{v} = \boxed{-7}\)