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Chapter 9: Problem 47

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

Expert verified
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

01

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.
02

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, \( \mathbf{u} \times \mathbf{v} = k\mathbf{v} \times\mathbf{v} \)
03

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.
04

Assume the converse

Assume that the cross product of the vectors \( \mathbf{u} \) and \( \mathbf{v} \) is the zero vector. \( \mathbf{u} \times \mathbf{v} = \mathbf{0} \).
05

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.

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