Precalculus · N-VM
Dot Product of Vectors — Magnitude and Angle Between Vectors
u·v = u₁v₁ + u₂v₂. Also u·v = |u||v|cos θ, so cos θ = (u·v)/(|u||v|). Perpendicular vectors have dot product 0.
PrecalculusCommon Core HSN-VM.B.3N-VMVerified correct ✓
Worked examples
Find the dot product of u = ⟨0, 3⟩ and v = ⟨-5, -4⟩. u · v = u₁v₁ + u₂v₂.
Answer: -12
Find the angle (in degrees) between u = ⟨-1, -2⟩ and v = ⟨1, 1⟩. Use cos θ = (u·v)/(|u||v|). Round to 2 decimal places.
Answer: 161.57
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