Question

Prove the property of the cross product. $$ \mathbf{u} \times \mathbf{v} \text { is orthogonal to both } \mathbf{u} \text { and } \mathbf{v} \text { . } $$

Short answer

By applying the definitions of cross product and dot product, and the principle of orthogonality, it has been proven that the cross product of any two vectors is orthogonal to both of those vectors.

Step-by-step solution

Definition of cross product

First, recall that the cross product of two vectors \( \mathbf{u} \) and \( \mathbf{v} \) in \( \mathbb{R}^3 \), denoted \( \mathbf{u} \times \mathbf{v} \), is another vector \( \mathbf{w} \) in \( \mathbb{R}^3 \) that is orthogonal to both \( \mathbf{u} \) and \( \mathbf{v} \), and whose magnitude is equal to the area of the parallelogram spanned by \( \mathbf{u} \) and \( \mathbf{v} \).

Orthogonality test

Then, proceed by proving that \( \mathbf{w}=\mathbf{u} \times \mathbf{v} \) is orthogonal to both \( \mathbf{u} \) and \( \mathbf{v} \). By definition, two non-null vectors are orthogonal if and only if their dot product is zero, i.e., \( \mathbf{a} \cdot \mathbf{b}=0 \). Therefore, for vector \( \mathbf{w} \) to be orthogonal to vectors \(\mathbf{u} \) and \( \mathbf{v} \), it must satisfy the conditions \( \mathbf{w} \cdot \mathbf{u} = 0 \) and \( \mathbf{w} \cdot \mathbf{v} = 0 \).

Apply distributive law

Verify these conditions. It's known that cross product is distributive over addition and dot product is distributive over addition. If factored out, we get \( \mathbf{w} \cdot \mathbf{u} = \mathbf{u} \times \mathbf{v} \cdot \mathbf{u} = \mathbf{u} \cdot \mathbf{u} \times \mathbf{v} = 0 \) and \( \mathbf{w} \cdot \mathbf{v} = \mathbf{u} \times \mathbf{v} \cdot \mathbf{v} = \mathbf{u} \cdot \mathbf{v} \times \mathbf{v} = 0 \). Therefore, \( \mathbf{u} \times \mathbf{v} \) is orthogonal to both \( \mathbf{u} \) and \( \mathbf{v} \).