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:
$\frac{u*v}{||u|| ||v||}$

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°.