Question

Show that if u and v are orthogonal unit vectors then $\mathit{u}\times \mathit{v}$is also a unit vector.

Short answer

To show that $\mathit{u}\times \mathit{v}$is a unit vector, first write for any two vectors and substitute the value of $\theta$in the given equation. As given in the question, substitute the values and solve.

Step-by-step solution

Step 1. Given information.

Consider the given question,

uand v are orthogonal unit vectors.

$\mathit{u}\times \mathit{v}$

For any two vectors $\left|\left|\mathit{u}\times \mathit{v}\right|\right|=\left|\left|\mathit{u}\right|\right|\left|\left|\mathit{v}\right|\right|\sin \theta$.

If uand v are orthogonal then $\theta =90°\mathit{u}\perp \mathit{v}$ , that is to say$\theta =90°$.

Step 2. Substitute the value of θ in the given equation.

Substitute the value of $\theta$in the given equation,

localid="1647011889263" $$\begin{gathered} \left|\left|\mathit{u}\times \mathit{v}\right|\right|=\left|\left|\mathit{u}\right|\right|\left|\left|\mathit{v}\right|\right|\sin 90° \\ \left|\left|\mathit{u}\times \mathit{v}\right|\right|=\left|\left|\mathit{u}\right|\right|\left|\left|\mathit{v}\right|\right|·1 \\ \left|\left|\mathit{u}\times \mathit{v}\right|\right|=\left|\left|\mathit{u}\right|\right|\left|\left|\mathit{v}\right|\right| \end{gathered}$$

As u and v are orthogonal unit vector, then know $$\begin{gathered} \left|\left|\mathit{u}\right|\right|=1, \\ \left|\left|\mathit{v}\right|\right|=1 \end{gathered}$$. Then,

$$\begin{gathered} \left|\left|\mathit{u}\times \mathit{v}\right|\right|=\left(1\right)\left(1\right) \\ \left|\left|\mathit{u}\times \mathit{v}\right|\right|=1 \end{gathered}$$

Hence, $\mathit{u}\times \mathit{v}$is a unit vector.