Dot product

Multiplying vectors to get a number

The dot product (also called the scalar product) of two vectors gives a scalar, not a vector.

Component formula:

u · v = u₁v₁ + u₂v₂ (2D)

u · v = u₁v₁ + u₂v₂ + u₃v₃ (3D)

Geometric formula:

u · v = |u||v| cos θ, where θ is the angle between them.

Examples

• [3, 4] · [2, −1] = 6 − 4 = 2
• [1, 2, 3] · [4, 0, −1] = 4 + 0 − 3 = 1

Finding the angle between vectors

cos θ = (u · v) / (|u||v|)

Example: u = [1, 2], v = [3, 1]

• u · v = 5
• |u| = √5, |v| = √10
• cos θ = 5/√50 ≈ 0.707 → θ = 45°

This is the angle between vectors.

What the sign tells you

Testing for perpendicular

Two non-zero vectors are orthogonal (perpendicular) if u · v = 0.

Example: Are [2, −3] and [6, 4] perpendicular? 12 − 12 = 0 → yes.

Find k: [k, 3] ⊥ [2, −4] → 2k − 12 = 0 → k = 6.

Properties

Application: work

In physics, work = force · displacement:

W = F · d = |F||d| cos θ

Example: Pulling a wagon 20 m with a 50 N force at 30° to the ground.

W = 50 × 20 × cos 30° ≈ 866 J

Only the part of the force in the direction of motion does work.

Application: projection

The dot product gives the projection of one vector onto another (see the projection lesson): scalar projection = (u · v)/|v|.

Worked examples

Example 1: [5, −2] · [1, 3] = −1 (obtuse angle).

Example 2: Angle between [1, 0] and [0, 1]? Dot = 0 → 90°.

Example 3: |v|² for v = [2, 3, 6]? v · v = 4 + 9 + 36 = 49.

Common mistakes

• Thinking the dot product is a vector.
• Forgetting to divide by both magnitudes when finding the angle.
• Multiplying the wrong components together.

Try it yourself

1. [4, 1] · [−2, 5]
2. [1, −1, 2] · [3, 3, 1]
3. Angle between [3, 4] and [4, 3]?
4. Find k so [k, 2, −1] ⊥ [3, k, 4].
5. Work done by F = [10, 5] N over d = [4, 0] m?

Answers: 1) −3 2) 2 3) ≈ 16.3° 4) k = 0.8 5) 40 J

Quick recap

• Dot product: u · v = sum of component products = |u||v| cos θ.
• Use it to find the angle between vectors; zero means orthogonal.
• Applications include work and projection.