The Dot Product Identity and the Cosine Rule in ℝ³

The Dot Product Identity and the Cosine Rule in ℝ³

In this article we derive the dot product identity

A · B = |A| × |B| × cos(θ)

and show how this identity leads directly to the cosine rule, using a combination of coordinate algebra and geometric interpretation.

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1. Vectors in ℝ³

Let the vectors be:

A = (a₁, a₂, a₃)
B = (b₁, b₂, b₃)

Their difference is:

A - B = (a₁ - b₁, a₂ - b₂, a₃ - b₃)

The squared magnitude of this difference vector is:

|A - B|² = (a₁ - b₁)² + (a₂ - b₂)² + (a₃ - b₃)².

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2. Expanding the Square of the Difference

Expand each component:

(a₁ - b₁)² = a₁² - 2a₁b₁ + b₁²
(a₂ - b₂)² = a₂² - 2a₂b₂ + b₂²
(a₃ - b₃)² = a₃² - 2a₃b₃ + b₃²

Adding these three expansions gives:

|A - B|² = (a₁² + a₂² + a₃²) + (b₁² + b₂² + b₃²) - 2(a₁b₁ + a₂b₂ + a₃b₃).

Recognise the squared magnitudes:

|A|² = a₁² + a₂² + a₃²
|B|² = b₁² + b₂² + b₃²

So the expression becomes:

|A - B|² = |A|² + |B|² - 2(a₁b₁ + a₂b₂ + a₃b₃).

Define the dot product:

A · B = a₁b₁ + a₂b₂ + a₃b₃.

This gives the algebraic identity:

|A - B|² = |A|² + |B|² - 2(A · B).

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3. Geometric Interpretation

To understand this identity geometrically, consider the vectors positioned so that each begins at the origin. Let θ be the angle between them.

The projection of B onto the direction of A has length:

|B| × cos(θ)

and the perpendicular component of B has length:

|B| × sin(θ).

Let the points representing vectors A and B be labelled A and B respectively. The distance between these endpoints is:

AB = |A - B|.

Along A’s direction, the horizontal displacement is:

AC = |A| - |B| cos(θ).

The perpendicular displacement is:

BC = |B| × sin(θ).

Triangle ABC is right-angled at C, so by the Pythagorean theorem:

AC² + BC² = AB².

Substituting the expressions for AC, BC, and AB:

(|A| - |B| cos(θ))² + (|B| sin(θ))² = |A - B|².

Expanding the left-hand side:

|A|² - 2|A||B| cos(θ) + |B|² cos²(θ) + |B|² sin²(θ).

Using the identity cos²(θ) + sin²(θ) = 1, this simplifies to:

|A - B|² = |A|² + |B|² - 2|A||B| cos(θ).

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4. Deriving the Dot Product Identity

We now have two expressions for |A - B|²:

(1)   |A|² + |B|² - 2(A · B)

(2)   |A|² + |B|² - 2|A||B| cos(θ)

Equating the middle terms gives:

A · B = |A| × |B| × cos(θ).

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5. The Cosine Rule in Vector Form

Substituting this identity into the expression for |A - B|² gives the cosine rule in vector form:

|A - B|² = |A|² + |B|² - 2|A||B| cos(θ).

This is the familiar cosine rule from triangle geometry, written entirely using vectors in ℝ³.