1. **Problem:** Calculate the dot product (also called scalar product) of the vectors given in part 1a: $\mathbf{u} = (2, -3)$ and $\mathbf{v} = (1, 2)$. 2. **Formula:** The dot product of two vectors $\mathbf{u} = (u_1, u_2, \ldots, u_n)$ and $\mathbf{v} = (v_1, v_2, \ldots, v_n)$ is given by $$\mathbf{u} \cdot \mathbf{v} = u_1 v_1 + u_2 v_2 + \cdots + u_n v_n$$ This means multiply corresponding components and sum the results. 3. **Calculation:** For $\mathbf{u} = (2, -3)$ and $\mathbf{v} = (1, 2)$, $$\mathbf{u} \cdot \mathbf{v} = 2 \times 1 + (-3) \times 2$$ Calculate each term: $$= 2 + (-6)$$ Simplify: $$= 2 - 6$$ $$= -4$$ 4. **Explanation:** The dot product is a single number that measures how much two vectors point in the same direction. Here, the result is $-4$, which is negative, indicating the vectors point somewhat in opposite directions. **Final answer:** $$\boxed{-4}$$