How do I find the dot product of $\overrightarrow{u}+\overrightarrow{v}$ and $\overrightarrow{u}-\overrightarrow{v}$ with the given information?

Question

I already know that $|\overrightarrow{u}|=2, |\overrightarrow{v}| = 3,|$ and $\langle\overrightarrow{u},\overrightarrow{v}\rangle = 1$. I am unsure as to how to proceed in order to find $\langle\overrightarrow{u}+\overrightarrow{v},\overrightarrow{u}-\overrightarrow{v}\rangle$. Can anyone help me here?

Answer 1

We have that $$(u+v) \cdot (u-v)=u \cdot (u-v)+v \cdot (u-v)= u \cdot u -u \cdot v + v \cdot u - v \cdot v =...$$ Can you finish it?

Answer 2

HINT: $(u+v,u-v) = (u,u) - (u,v) + (v,u) - (v,v)$. You know all of the terms of the RHS of the previous equation.

Answer 3

We have $$(\vec{u}+\vec{v})\cdot (\vec{u}-\vec{v})=\vec{u}^2-\vec{v}^2=|\vec{u}|^2-|\vec{u}|^2=…$$