gen-define-the-cross-product-and-give-one-short-worked-example-b686

Define the cross product and give one short worked example.

# Cross Product ## Definition The **cross product** of two vectors **a** and **b** in ℝ³ is a vector perpendicular to both, defined as: **a** × **b** = |**a**| |**b**| sin(θ) **n** where θ is the angle between the vectors and **n** is the unit vector perpendicular to both (determined by the right-hand rule). ## Component Formula For **a** = ⟨a₁, a₂, a₃⟩ and **b** = ⟨b₁, b₂, b₃⟩: **a** × **b** = ⟨a₂b₃ - a₃b₂, a₃b₁ - a₁b₃, a₁b₂ - a₂b₁⟩ ## Worked Example Find **a** × **b** where **a** = ⟨1, 2, 3⟩ and **b** = ⟨4, 5, 6⟩. **Solution:** **a** × **b** = ⟨(2)(6) - (3)(5), (3)(4) - (1)(6), (1)(5) - (2)(4)⟩ = ⟨12 - 15, 12 - 6, 5 - 8⟩ = ⟨-3, 6, -3⟩ **Answer:** **a** × **b** = ⟨-3, 6, -3⟩ (You can verify: this result is perpendicular to both original vectors since ⟨-3, 6, -3⟩ · ⟨1, 2, 3⟩ = -3 + 12 - 9 = 0 ✓)