Convention: A letter followed by an underscore means a vector: A_
NOTE: Angle between vectors is always between 0º and 180º, inclusive.
A_•B_ = AB cosθ = -7
|A_x B_| = AB sinθ = 6
Sum of their squares is:
(AB cosθ)^2 + (AB sinθ)^2 = -7^2 + 6^2 = 85
= (AB)^2 ((cosθ)^2 + (sinθ)^2) = (AB)^2
and since
A, B = |A_|, |B_| ≥ 0,
AB = sqrt(85) (and not -10)
So
sqrt(85) cosθ = -7; sqrt(85) sinθ = 6
(cosθ, sinθ) = (-0.76, 0.65)
θ = acos(-0.76) = 139.46 degrees
Therefore, the angle between the two vectors is 139.46 degrees.
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