Anticommutative property

Definition

The anticommutative property states that the order of the operands affects the sign of the result. Specifically, for vectors, $\mathbf{a} \times \mathbf{b} = - (\mathbf{b} \times \mathbf{a})$.

5 Must Know Facts For Your Next Test

1. The anticommutative property is essential when dealing with the cross product of two vectors.
2. It implies that swapping the vectors in a cross product changes the direction of the resulting vector to its opposite.
3. The property is used to determine torque and angular momentum calculations in physics.
4. Understanding this property helps in solving problems related to rotational dynamics and forces in three dimensions.
5. It highlights that not all vector operations are commutative, contrasting with dot products which are commutative.

Review Questions

• What does the anticommutative property imply about the cross product of two vectors?
• How does changing the order of vectors in a cross product affect the result?
• Give an example where the anticommutative property is applied in physics.

Related terms

Cross Product: A binary operation on two vectors in three-dimensional space resulting in another vector perpendicular to both.

Dot Product: A scalar value obtained from multiplying two equal-length sequences of numbers and summing their products.

Torque: $\mathbf{\tau} = \mathbf{r} \times \mathbf{F}$; A measure of how much a force acting on an object causes that object to rotate.