How do you find the angle between vectors?

Question

I fell asleep while she was teaching it and now i have like 3 pages due tommorrow. Any help help would be great. ex. find the angle between the vectors when u=<-4,-3>, v=<-1,5> if you could explain how to do this i would appreciate alot. thank you

Answer 1

Given two vectors

u = <-4, -3>
v = <-1, 5>

find the angle θ between them.



Use the dot product. By definition we have:

u • v = | u | | v | cosθ

cosθ = (u • v) / (| u | | v |)

cosθ = (<-4, -3> • <-1, 5>) / {√[(-4)² + (-3)²] * √[(-1)² + 5²]}

cosθ = [(-4)*(-1) + (-3)(5)] / {√[16 + 9] * √[1 + 25]}

cosθ = [4 - 15] / {√25 * √26} = -11/(5√26)

θ = arccos[-11/(5√26)] ≈ 115.55997°

Answer 2

The angle between u and v is given by θ = arccos (u·v/(|u|*|v|)). So:

u·v = (-4)*(-1) + (-3)*5 = 4-15 = -11
|u| = √((-4)² + (-3)²) = √25 = 5
|v| = √((-1)² + 5²) = √26

θ = arccos (-11/(5√26)) ≈ 115.56°