Parametric, Polar & Vectors · Vector Operations

Cross Product

\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).

6 variables
2 worked examples
12 in Parametric, Polar & Vectors

Variables

Variables used in the Cross Product formula
Symbol	Name	Unit
u1	u x-component	-
u2	u y-component	-
u3	u z-component	-
v1	v x-component	-
v2	v y-component	-
v3	v z-component	-

Worked examples

Find ⟨1, 2, 3⟩ × ⟨4, 5, 6⟩.

i: (2)(6) - (3)(5) = 12 - 15 = -3
j: -[(1)(6) - (3)(4)] = -(6 - 12) = 6
k: (1)(5) - (2)(4) = 5 - 8 = -3

Answer: ⟨-3, 6, -3⟩

1 more worked examplePremium

Find ⟨2, 0, -1⟩ × ⟨1, 3, 4⟩.

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