Key Idea: The scalar (dot) product turns two vectors into one number. It finds angles and tests for perpendicular vectors. It is on Paper 1 and Paper 2.
Dot product & angle between vectors
- Components: v·w = v₁w₁ + v₂w₂ + v₃w₃.
- Angle: cos θ = (v·w)/(|v||w|), then cos⁻¹.
- The answer is a number, never a vector.
Perpendicular: 90°. v·w = 0.
Parallel: 0° or 180°. v = t w (same ratio).
A zero dot product means a right angle.
cos θ for a = (n, 1) and b = (1, n)? 2n/(n² + 1).
What does cos θ approach as n gets very large? 0, so θ approaches 90°.
Is (2, 4) parallel to (1, 2)? Yes: (2, 4) = 2(1, 2).
OM·MC = −2.5. Are they perpendicular? No, the dot product is not 0.