PROOF 156

Prove.

The dot product is the product of magnitudes times cos

On plane vectors.

What you may use

A1c² = a²+b²−2ab cos C
law of cosines (Proof 34)
A2PQ = √((Δx)²+(Δy)²)
distance between two points (Proof 21)
A3x ≥ 0 ⟹ (√x)² = x
definition of the square root
A4(u−v)² = u²−2uv+v²
expansion formula (Proof 23)
A5θ ∈ {0°,180°} ⟺ b = ta
when the two vectors lie on one line — 0° if t > 0, 180° if t < 0
A6cos 0° = 1, cos 180° = −1
cosine of special angles
A7√(xy) = √x·√y (x,y≥0)
square root of a product (Proof 96)
A8√(t²) = |t|
square root and absolute value (Proof 97)
A9|t| = t or |t| = −t
definition of absolute value — t if t ≥ 0, −t if t < 0
A10a(b + c) = ab + ac
distributive law
A11a + b = b + a, ab = ba
commutativity of addition and multiplication
A12(a+b)+c = a+(b+c)
associativity of addition
A13(ab)c = a(bc)
associativity of multiplication
A14c≠0, cx = cy ⟹ x = y
cancellation law
Ea = b ⟹ same operation on both sides
properties of equality

Nothing outside this list may be cited.

Definition D1

For two plane vectors a = (a₁, a₂), b = (b₁, b₂), define a·b = a₁b₁ + a₂b₂.

Proof.

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