Question

If $\overrightarrow{a,}\overrightarrow{b,}\overrightarrow{c}$are three unit vectors such that $\overrightarrow{a}\times \overrightarrow{b}=\overrightarrow{c},\overrightarrow{b}\times \overrightarrow{c}=\overrightarrow{a,}\overrightarrow{c}\times \overrightarrow{a}=\overrightarrow{b}.$ Show that $\overrightarrow{a,}\overrightarrow{b,}\overrightarrow{c}$form an orthonormal right handed triad of unit vectors.

Answer 1

$$\begin{gathered} \text{Given:} \\ \overrightarrow{a}\times \overrightarrow{b}=\overrightarrow{c} \\ \overrightarrow{b}\times \overrightarrow{c}=\overrightarrow{a} \\ \overrightarrow{c}\times \overrightarrow{a}=\overrightarrow{b}...\left(1\right) \\ \text{Now,} \\ \left|\overrightarrow{a}\times \overrightarrow{b}\right|=\left|\overrightarrow{c}\right|=1\text{(∵}\overrightarrow{c}\text{is a unit vector)} \\ \left|\overrightarrow{b}\times \overrightarrow{c}\right|=\left|\overrightarrow{a}\right|=1\text{(∵}\overrightarrow{a}\text{is a unit vector)} \\ \left|\overrightarrow{c}\times \overrightarrow{a}\right|=\left|\overrightarrow{b}\right|=1\text{(∵}\overrightarrow{b}\text{is a unit vector)} \\ \therefore \left|\overrightarrow{a}\times \overrightarrow{b}\right|=\left|\overrightarrow{b}\times \overrightarrow{c}\right|=\left|\overrightarrow{c}\times \overrightarrow{a}\right|=1...\left(2\right) \\ \text{From (1) and (2), we know} \\ \overrightarrow{a},\overrightarrow{b}\text{and}\overrightarrow{c}\text{form an orthonormal right handed triad of unit vectors.} \end{gathered}$$