Question
Prove that $\langle a \mathbf{v}+b \mathbf{w}, c \mathbf{v}+d \mathbf{w}\rangle=a c\|\mathbf{v}\|^2+(a d+b c)\langle\mathbf{v}, \mathbf{w}\rangle+b d\|\mathbf{w}\|^2$.
Step 1
The dot product of two vectors $\mathbf{u}$ and $\mathbf{v}$ is given by $\langle \mathbf{u}, \mathbf{v} \rangle$. We have: \[ \langle a \mathbf{v} + b \mathbf{w}, c \mathbf{v} + d \mathbf{w} \rangle \] Show more…
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