AHL 3.13 — THE SCALAR (DOT) PRODUCT OF VECTORS

The scalar product is a way to multiply two vectors and obtain a single real number.
It connects the algebraic world (components of vectors) and the geometric world (angles between directions).
It is one of the most important tools in higher mathematics, physics, engineering, computer graphics, and AI.

Concept Explanation
Algebraic definition of the dot product If v = (v1, v2, v3) and w = (w1, w2, w3), then:v · w = v1 × w1 + v2 × w2 + v3 × w3

This produces a real number, not a vector.
Geometric definition The dot product also measures how much two vectors “align”:v · w = |v| × |w| × cosθ
where θ is the angle between the vectors.

If the vectors point in the same direction → cosθ = 1 → large positive product.
If the vectors are perpendicular → cosθ = 0 → product = 0.
If vectors point in opposite directions → cosθ = −1 → negative product.
Perpendicular vectors Two vectors are perpendicular (orthogonal) exactly when:v · w = 0

Reason: if θ = 90°, then cos90° = 0 → dot product becomes 0.

This is one of the most important uses of the dot product and appears in geometry, physics, and coding.
Parallel vectors Vectors are parallel when one is a scalar multiple of the other:w = k × v for some scalar k

Then:

v · w = |v| × |w| for k > 0
v · w = −|v| × |w| for k < 0

📌 Why does v · w = |v| × |w| × cosθ make sense?

The expression |v| × cosθ represents the length of the projection of v onto w.
Multiplying by |w| scales this projection relative to the length of w, creating a measurement of “how much of v goes in the direction of w”.

Thus, the dot product connects the geometry of angles and projections with the algebra of vector components.

📌 Properties of the Dot Product

  • v · w = w · v (commutative)
  • v · (w + u) = v · w + v · u (distributive)
  • (k × v) · w = k(v · w) (scalar multiplication)
  • v · v = |v|2 (useful shortcut)
  • v · w = 0 ↔ vectors are perpendicular

Example 1 — Dot Product

v = (3, −1, 4), w = (2, 5, −1)

v · w = (3 × 2) + (−1 × 5) + (4 × −1)
= 6 − 5 − 4
= −3

Since the result is negative → θ > 90°, the vectors point in broadly opposite directions.

Example 2 — Perpendicular Checking

v = (3, 1, −2), w = (2, −6, −1)

v · w = (3 × 2) + (1 × −6) + (−2 × −1)
= 6 − 6 + 2
= 2

Not perpendicular, because perpendicular vectors produce a dot product of exactly 0.

Example 3 — Angle Between Vectors

Use θ = arccos((v · w) ÷ (|v| × |w|)).

This allows angles to be computed given only coordinates.

🌍 Real-World Applications

  • Physics: work = force · displacement
  • Robotics: measuring alignment of velocity and direction
  • Computer graphics: lighting models use n · L (normal · light direction)
  • Aviation: angle between wind and flight path
  • Machine learning: cosine similarity uses dot product to compare vectors of data

📐 IA Ideas

  • Analyse how dot product determines efficiency in rowing, swimming, cycling strokes.
  • Modelling how much a force contributes to displacement in mechanics.
  • Using cosine similarity to compare text vectors in NLP.
  • Investigate the dot product in 3D geometry (intersections, heights, projections).

🌐 EE Connections

  • Deep investigation of Euclidean vs non-Euclidean inner products.
  • Connections between dot products, projections, and orthogonality in abstract vector spaces.
  • Mathematical analysis of machine-learning similarity metrics.

❤️ CAS Ideas

  • Tutoring younger students on force components using dot products.
  • Helping physics classes calculate work done in experiments.
  • Building physical models to demonstrate perpendicular vs parallel vectors.

📱 GDC Use

  • Store vectors in memory: vec1, vec2
  • Use the vector dot-product command for quick checking
  • Compute |v| using the magnitude command
  • Compute θ using:
    acos((v·w) ÷ (|v| × |w|))
  • Check perpendicularity: if the calculator returns 0, vectors are orthogonal

🔍 TOK Perspective

  • How does turning geometry into algebra change mathematical understanding?
  • Is the dot product a discovery (from nature) or an invention (a tool)?
  • Why do we define perpendicularity through the number 0?

📝 Examiner Tips

  • Always check if vectors are perpendicular by testing v · w = 0.
  • Use v · v = |v|2 to simplify calculations.
  • Be careful with negative signs in component multiplication.
  • For angle problems, simplify the fraction before applying arccos.