PROOF 157

Prove.

If the dot product is zero, the vectors are perpendicular

On plane vectors.

A way to read this proof

First separate what is assumed from what is to be proved. At each step, ask which definition or earlier result permits the move. If it is unclear, return to that premise rather than skipping to the conclusion.

Definition D1

Call 0 = (0, 0) the zero vector. When a ≠ 0, b ≠ 0 and the angle between the two vectors is 90°, write a ⊥ b and call the two vectors perpendicular.

Proof.

(1)The angle θ between two vectors is the angle between the two rays drawn from the origin, and 0° ≤ θ ≤ 180°.convention

(2)The zero vector determines no ray, so θ is defined only when neither vector is the zero vector. Perpendicularity has meaning only then.(1), convention

∎

Lemma L1

If a ≠ 0, b ≠ 0 and θ = 90°, then a·b = 0

Proof.

(1)θ = 90°, so cos θ = 0assumption, A4

(2)a·b = |a||b| cos θ = (|a||b|) × 0 = 0(1), A1, A9, A6, E

∎

Lemma L2

If a ≠ 0, b ≠ 0 and a·b = 0, then θ = 90°

Proof.

(1)|a| > 0 and |b| > 0, so |a| ≠ 0, |b| ≠ 0assumption, A3, A8

(2)0 = a·b = |a||b| cos θ = |a| (|b| cos θ)assumption, A1, A9, E

(3)|a| ≠ 0, so |b| cos θ = 0(1), (2), A5

(4)|b| ≠ 0, so cos θ = 0(1), (3), A5

(5)0° ≤ θ ≤ 180°, so θ = 90°(4), A4, D1

∎

Theorem T

For a ≠ 0, b ≠ 0, a·b = 0 and a ⊥ b are the same.

Proof.

(1)a ⊥ b means θ = 90°.D1

(2)If θ = 90°, then a·b = 0L1

(3)If a·b = 0, then θ = 90°L2

(4)Both directions hold, so a·b = 0 ⟺ a ⊥ b(1), (2), (3)

∎

Corollary C

In theorem T, the assumption a ≠ 0, b ≠ 0 cannot be dropped.

Proof.

(1)For any vector b, 0·b = 0 × b₁ + 0 × b₂ = 0 + 0 = 0A2, A6, A7

(2)The zero vector determines no ray, so the angle between 0 and b is undefined, and 0 ⊥ b has no meaning.D1

(3)Dropping the assumption, for every pair in (1) the left side is true and the right side has no meaning, so the theorem does not survive as stated.(1), (2), T

(4)That the dot product is 0 means the vectors are perpendicular, or one of them is the zero vector.(1), (3), T

∎

Remark

Can A1 be removed from this proof?

Keep only the component formula, and check what the computation a₁b₁ + a₂b₂ = 0 says about the angle.