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3-3 (g) Definition of the vector product

3-3 (g) Definition of the vector product

Point 1 of 7 — The Vector (Cross) Product and the Scalar Triple ProductQuestion bank: 0 questions (0 with pictures)

01 3-3 (g) Definition of the vector product

Definition

  • Let A⃗ and B⃗ be two vectors in space with angle θ between them (0∘≤θ≤180∘).
  • Their vector product (cross product) is A⃗×B⃗=(∥A⃗∥ ∥B⃗∥sin⁡θ)n^
  • n^ is the unit vector perpendicular to the plane containing A⃗ and B⃗.
  • The result is a vector, in contrast with the scalar product which is a number.

Direction: the right-hand rule

  • Curl the fingers of the right hand from the first vector A⃗ towards the second vector B⃗ through the smaller angle.
  • The thumb points along n^, the direction of A⃗×B⃗.
  • Swapping the order reverses the direction: B⃗×A⃗=−(A⃗×B⃗).

The unit vectors

  • For a right-handed system: i^×j^=k^, j^×k^=i^, k^×i^=j^.
  • In the reverse order the sign changes: j^×i^=−k^, k^×j^=−i^, i^×k^=−j^.
  • i^×i^=j^×j^=k^×k^=0⃗.

Norm of the product

  • ∥A⃗×B⃗∥=∥A⃗∥ ∥B⃗∥sin⁡θ.
  • It is zero when the vectors are parallel (θ=0∘ or 180∘) and largest when they are perpendicular.
✍️ Worked examples

Example 1: Two vectors in a plane have norms 15 and 17.5 and the angle between them is 70∘. Find ∥A⃗×B⃗∥.

Solution:

  • ∥A⃗×B⃗∥=15×17.5×sin⁡70∘=262.5×0.9397≈246.7.

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