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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).

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

Variables

Variables used in the Cross Product formula
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⟩

1 more worked examplePremium

Find ⟨2, 0, -1⟩ × ⟨1, 3, 4⟩.

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Common questions

What is the Cross Product?

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). It is one of the parametric, polar & vectors formulas in the CalcRef reference.

What do the symbols in the Cross Product mean?

In the Cross Product, u1 is the u x-component; u2 is the u y-component; u3 is the u z-component; v1 is the v x-component; v2 is the v y-component; v3 is the v z-component.

How do you use the Cross Product?

Worked example. 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⟩.

Practice

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