Question

If
`v=v_(1) +v_(2)`, where
`v_(1)` and
`v_(2)` are not both 0 and
`u=-3v`, find the measure of the angle between
`u` and
`v`.

Given:

`(u*v)/(||u|| ||v||)`

Answer 1

If θ is the angle between u and v, then

u • v = ||u|| ||v|| cos(θ)

Since u = -3v, we have

u • v = (-3v) • v = -3 ||v||²

and

||u|| = ||-3v|| = 3 ||v||

If v₁ and v₂ are both not zero, then ||v| ≠ 0.

Then

cos(θ) = (-3 ||v||²) / (3 ||v|| ||v||) = 1

which means θ = π rad = 180°.