Question

Under what conditions is \(\mathbf{u} \times \mathbf{v}\) a unit vector?

Short answer

Answer: The cross product of two vectors will result in a unit vector when the product of the magnitudes of the vectors and the sine of the angle between them is equal to 1, i.e., \(|\mathbf{u}| \cdot |\mathbf{v}| \cdot \sin{(\theta)} = 1\).

Step-by-step solution

Define the cross product

The cross product of two vectors \(\mathbf{u}\) and \(\mathbf{v}\) is a vector \(\mathbf{w}\) where: \(\mathbf{w} = \mathbf{u} \times \mathbf{v}\). The magnitude of the cross product is given by: \(|\mathbf{w}| = |\mathbf{u}| \cdot |\mathbf{v}| \cdot \sin{(\theta)}\), where \(|\mathbf{u}|\) and \(|\mathbf{v}|\) are the magnitudes of vectors \(\mathbf{u}\) and \(\mathbf{v}\), and \(\theta\) is the angle between them.

Find the condition for a unit vector

A unit vector is a vector with a magnitude of 1. So, to find the condition under which the cross product of \(\mathbf{u}\) and \(\mathbf{v}\) is a unit vector, we need to find when \(|\mathbf{w}| = 1\). Using the formula for the magnitude of the cross product, we get: \(1 = |\mathbf{u}| \cdot |\mathbf{v}| \cdot \sin{(\theta)}\).

Obtain the condition

To satisfy the equation \(1 = |\mathbf{u}| \cdot |\mathbf{v}| \cdot \sin{(\theta)}\), we need the following condition to hold true: \(|\mathbf{u}| \cdot |\mathbf{v}| \cdot \sin{(\theta)} = 1\). In summary, under the condition that the product of the magnitudes of the vectors \(\mathbf{u}\) and \(\mathbf{v}\) and the sine of the angle between them (\(\theta\)) is equal to 1, the cross product of the vectors \(\mathbf{u}\) and \(\mathbf{v}\) will result in a unit vector.