Question

Prove the property of the cross product. \(\mathbf{u} \times \mathbf{v}=\mathbf{0}\) if and only if \(\mathbf{u}\) and \(\mathbf{v}\) are scalar multiples of each other.

Short answer

Thus, the property has been proven: The cross product of two vectors is the zero vector if and only if the vectors are scalar multiples of each other.

Step-by-step solution

Assume the truth of the property

To commence, assume that the given vectors \( \mathbf{u} \) and \( \mathbf{v} \) are scalar multiples of each other. This could be represented as: \( \mathbf{u} = k\mathbf{v} \), where k is a scalar multiplier.

Calculate the cross product

Now calculate the cross product of these vectors. Remember that cross product of two vectors \( \mathbf{A} \) and \( \mathbf{B} \), denoted by \( \mathbf{A} \times \mathbf{B} \), is defined in such a way that: \( \mathbf{A} \times \mathbf{B} = - \mathbf{B} \times \mathbf{A} \). Thus, \( \mathbf{u} \times \mathbf{v} = k\mathbf{v} \times\mathbf{v} \)

Apply the property of cross products

It is a known property that any vector crossed with itself results in the zero vector because the angle between them is zero degree. Therefore, \( \mathbf{u} \times \mathbf{v} = k\mathbf{0} \) which further simplifies to \( \mathbf{u} \times \mathbf{v} = \mathbf{0} \). This verifies one direction of the proof.

Assume the converse

Assume that the cross product of the vectors \( \mathbf{u} \) and \( \mathbf{v} \) is the zero vector. \( \mathbf{u} \times \mathbf{v} = \mathbf{0} \).

Show vectors are scalar multiples

The zero cross product vector indicates that the two vectors are parallel and hence scalar multiples. This completes the proof.