Question

Show that if u and v are orthogonal then

$\left|\left|\mathit{u}\times \mathit{v}\right|\right|=\left|\left|\mathit{u}\right|\right|\left|\left|\mathit{v}\right|\right|$

Short answer

To show $\left|\left|\mathit{u}\times \mathit{v}\right|\right|=\left|\left|\mathit{u}\right|\right|\left|\left|\mathit{v}\right|\right|$, first write for any two vectors and substitute the value of$\theta$in the given equation and solve.

Step-by-step solution

Step 1. Given information.

Consider the given question,

$\left|\left|\mathit{u}\times \mathit{v}\right|\right|=\left|\left|\mathit{u}\right|\right|\left|\left|\mathit{v}\right|\right|$

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,

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

Therefore, $LHS=RHS$.

Hence,$\left|\left|\mathit{u}\times \mathit{v}\right|\right|=\left|\left|\mathit{u}\right|\right|\left|\left|\mathit{v}\right|\right|$proved.