To answer this question, let's break down the information:

1. We are given two non-zero vectors $\mathbf{u}$ and $\mathbf{v}$.
2. The condition provided is: $\mathbf{u} \times \mathbf{v} = \mathbf{v} \times \mathbf{u}$
3. We need to determine if this condition implies that $\mathbf{u}$ and $\mathbf{v}$ are parallel.

Analysis:

The cross product of two vectors $\mathbf{u}$ and $\mathbf{v}$ is defined as: $\mathbf{u} \times \mathbf{v} = -(\mathbf{v} \times \mathbf{u})$ which means the cross product is anti-commutative. Therefore, if $\mathbf{u} \times \mathbf{v} = \mathbf{v} \times \mathbf{u}$, then: $\mathbf{u} \times \mathbf{v} = -(\mathbf{u} \times \mathbf{v}) \Rightarrow \mathbf{u} \times \mathbf{v} = \mathbf{0}$ This result, $\mathbf{u} \times \mathbf{v} = \mathbf{0}$, implies that $\mathbf{u}$ and $\mathbf{v}$ are either parallel or one of them is the zero vector. Since we are given that neither $\mathbf{u}$ nor $\mathbf{v}$ is zero, the only possibility is that $\mathbf{u}$ and $\mathbf{v}$ are parallel.

Conclusion:

The statement is True. If $\mathbf{u} \times \mathbf{v} = \mathbf{v} \times \mathbf{u}$, then $\mathbf{u}$ and $\mathbf{v}$ are indeed parallel.

Would you like further details on the cross product properties, or do you have any additional questions?

Related Questions

1. What does it mean for two vectors to be parallel in 3D space?
2. How can you determine the direction of the cross product between two vectors?
3. What are the properties of the cross product of two vectors?
4. How can you prove that the cross product of two parallel vectors is zero?
5. Why is the cross product anti-commutative?

Tip

Remember that if the cross product of two non-zero vectors is zero, it always indicates that the vectors are parallel.