Given $ ||\vec{u}|| = 2$, $||\vec{v}|| = 3$, and $\vec{u} \cdot \vec{v} = -1$, find $(\vec{u} + \vec{v}) \cdot (2 \vec{u} - \vec{v})$.

Question

Given $ ||\vec{u}|| = 2$, $||\vec{v}|| = 3$, and $\vec{u} \cdot \vec{v} = -1$, find $(\vec{u} + \vec{v}) \cdot (2 \vec{u} - \vec{v})$.

Do I need to find what $\vec{u}$ and $\vec{v}$ actually are to answer this question or is there another way?

Answer 1

Hint: Expanding we get $$2\vec{u}^2+2\vec{u}\cdot \vec{v}-\vec{u}\cdot \vec{v}-\vec{v}^2=4-2-9=4-11=-7$$

Answer 2

Hint: You just have to use these facts:

• $\vec u.\vec u=4$:
• $\vec v.\vec v=9$;
• $\vec u.\vec v=-1$.