Cross Product Math Example 4

Follow the full solution, then compare it with the other examples linked below.

Example 4

easy
Is a×b=b×a\mathbf{a} \times \mathbf{b} = \mathbf{b} \times \mathbf{a}?

Solution

  1. 1
    No. a×b=−(b×a)\mathbf{a} \times \mathbf{b} = -(\mathbf{b} \times \mathbf{a}).
  2. 2
    The cross product is anti-commutative — reversing the order negates the result.

Answer

No, a×b=−(b×a)\mathbf{a} \times \mathbf{b} = -(\mathbf{b} \times \mathbf{a}).
Unlike the dot product, the cross product is anti-commutative. Swapping the operands reverses the direction of the resulting vector. This reflects the right-hand rule: the direction depends on order.

About Cross Product

The cross product of two 3D vectors a=⟨a1,a2,a3⟩\mathbf{a} = \langle a_1, a_2, a_3 \rangle and b=⟨b1,b2,b3⟩\mathbf{b} = \langle b_1, b_2, b_3 \rangle is a new vector a×b\mathbf{a} \times \mathbf{b} that is perpendicular to both a\mathbf{a} and b\mathbf{b}. Its magnitude equals the area of the parallelogram formed by a\mathbf{a} and b\mathbf{b}.

Learn more about Cross Product →

More Cross Product Examples

Example 1 medium

Find [formula].

Example 2 hard

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Example 3 medium

Find [formula].

← All Cross Product Examples Cross Product Concept