Question

What is the dot product $(\mathbf{u}-2 \mathbf{v}) \cdot(3 \mathbf{u}+4 \mathbf{v})$

Answer 1

\begin{aligned}(\mathbf{u}-2 \mathbf{v}) \cdot(3 \mathbf{u}+4 \mathbf{v}) &=\mathbf{u} \cdot(3 \mathbf{u}+4 \mathbf{v})-2 \mathbf{v} \cdot(3 \mathbf{u}+4 \mathbf{v}) \\ &=3(\mathbf{u} \cdot \mathbf{u})+4(\mathbf{u} \cdot \mathbf{v})-6(\mathbf{v} \cdot \mathbf{u})-8(\mathbf{v} \cdot \mathbf{v}) \\ &=3\|\mathbf{u}\|^{2}-2(\mathbf{u} \cdot \mathbf{v})-8\|\mathbf{v}\|^{2} \end{aligned}