The Scalar Product (Dot Product)

This lesson covers the scalar product (also called the dot product) of two vectors — as required by the Edexcel A-Level Mathematics specification (9MA0). You need to know both the geometric and component forms, use it to find angles between vectors, and apply the condition for perpendicularity.

---

Definition — Geometric Form

The scalar product of two vectors a and b is defined as:

a · b = |a| |b| cos θ

where θ is the angle between the vectors (when placed tail-to-tail), and 0° ≤ θ ≤ 180°.

Key Properties of the Geometric Form

• The result is a scalar (a number), not a vector.
• If θ = 90° (vectors are perpendicular), then a · b = 0 (since cos 90° = 0).
• If θ = 0° (vectors are parallel, same direction), then a · b = |a||b|.
• If θ = 180° (vectors are anti-parallel), then a · b = -|a||b|.

---

Definition — Component Form

For vectors a = (a₁, a₂, a₃) and b = (b₁, b₂, b₃):

a · b = a₁b₁ + a₂b₂ + a₃b₃