CalcRef

Parametric, Polar & Vectors · Vector Operations

Cross Product

u⃗×v⃗=∣i^j^k^u1u2u3v1v2v3∣\vec{u} \times \vec{v} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ u_1 & u_2 & u_3 \\ v_1 & v_2 & v_3 \end{vmatrix}

The cross product produces a vector perpendicular to both u and v. Its magnitude equals |u||v|sin θ (the area of the parallelogram formed by u and v).

Variables

SymbolNameUnit
u1u x-component—
u2u y-component—
u3u z-component—
v1v x-component—
v2v y-component—
v3v z-component—

Worked examples

Find ⟨1, 2, 3⟩ × ⟨4, 5, 6⟩.
  1. i: (2)(6) - (3)(5) = 12 - 15 = -3
  2. j: -[(1)(6) - (3)(4)] = -(6 - 12) = 6
  3. k: (1)(5) - (2)(4) = 5 - 8 = -3

Answer: ⟨-3, 6, -3⟩

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