Question

Prove \(\|\mathbf{u} \times \mathbf{v}\|=\|\mathbf{u}\|\|\mathbf{v}\|\) if \(\mathbf{u}\) and \(\mathbf{v}\) are orthogonal.

Short answer

The magnitude of the cross product of two orthogonal vectors \(\mathbf{u}\) and \(\mathbf{v}\) equals the product of their magnitudes, as defined by the relation \(\|\mathbf{u} \times \mathbf{v}\| = \|\mathbf{u}\|\|\mathbf{v}\|\).

Step-by-step solution

Define the terms

First, let's make sure we understand the notation: \(\mathbf{u}\) and \(\mathbf{v}\) are vectors; \(\|\mathbf{u}\|\) and \(\|\mathbf{v}\|\) represent their magnitudes; \(\mathbf{u} \times \mathbf{v}\) is their cross product; and the two-vectors are orthogonal, which means their dot product is zero.

Apply the formula for the magnitude of the cross product

The formula for the magnitude (or length) of the cross product of two vectors is given by \(\|\mathbf{u} \times \mathbf{v}\| = \|\mathbf{u}\|\|\mathbf{v}\|\sin(\theta)\), where \(\theta\) is the angle between \(\mathbf{u}\) and \(\mathbf{v}\).

Utilize the definition of orthogonality

The vectors \(\mathbf{u}\) and \(\mathbf{v}\) are orthogonal, which means \(\theta\) equals 90 degrees. The sine of 90 degrees is 1.

Substitution

Substitute \(\sin(90^\circ)\) with 1 in the formula from Step 2. The formula then simplifies to \(\|\mathbf{u} \times \mathbf{v}\| = \|\mathbf{u}\|\|\mathbf{v}\|\), thereby proving the given property.