Vectors $a$ and $b$ each have length $6$ and form an angle of $45°$ . Let $u = a + 2b$ and $v = 2a - b$. What is the angle formed by $u$ and $v$?

Question

Suppose vectors $a$ and $b$ each have length $6$ and form an angle of $45°$ . Let $u = a + 2b$ and $v = 2a - b$. What is the angle formed by $u$ and $v$ ?

I've drawn vector a and b on a coordinate plane with vector a being on the $x$ axis and b in quadrant $1$

I was wondering if it works to do $\cos(angle)=\frac{u\cdot v}{||v||||u||}$ and treat $u$ and $v$ as separate vectors?

I wasn't too sure on how to proceed from here

/btw a hint that was given to me was something about a dot product, I was wondering how I would use a dot product in this problem?/

Answer 1

HINT

$$u\cdot v= (a+2b)\cdot (2a-b)=2|a|^2-2|b|^2+3a\cdot b$$

and

$$a\cdot b=|a||b|\cos 45°$$

also

$$u\cdot v=|u||v|\cos \alpha$$

$$|u|^2=u\cdot u=(a+2b)(a+2b)=|a|^2+2|b|^2+4a\cdot b$$

Can you finish it?