Parametric, Polar & Vectors · Vector Operations

Dot Product

\vec{u} \cdot \vec{v} = u_1 v_1 + u_2 v_2 + u_3 v_3 = |\vec{u}||\vec{v}|\cos\theta

The dot product is a scalar equal to the sum of component products. It also equals the product of magnitudes times the cosine of the angle between them.

Variables

Symbol	Name	Unit
u1	u x-component	—
u2	u y-component	—
v1	v x-component	—
v2	v y-component	—

Worked examples

Find ⟨1, 2, 3⟩ · ⟨4, -5, 6⟩.

(1)(4) + (2)(-5) + (3)(6) = 4 - 10 + 18 = 12

Answer: 12

Are ⟨1, 2⟩ and ⟨-2, 1⟩ perpendicular?

(1)(-2) + (2)(1) = -2 + 2 = 0. Dot product is 0.

Answer: Yes, they are perpendicular.

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