Question

If $\overline{c}=\overline{3a}-\overline{2b}$, then prove that $\left[\overline{a}\overline{b}\overline{c}\right]=0$

Answer 1

Given that,

$\overline{c}=3\overline{a}-2\overline{b}$

To prove;$\left[\overline{a}\overline{b}\overline{c}\right]=0$

$\overline{b}\times \overline{b}=0$

In a scalar triple product , two vector are equal then the scalar triple product is $Zero$.

$L.H.S=$

$\begin{array}{c}=\left[\overline{a}\overline{b}\overline{c}\right]\\ =\overline{a}\left(\overline{b}\times \left(3\overline{a}-2\overline{b}\right)\right)\\ =\overline{a}\left(3\overline{a}\times \overline{b}-2\overline{b}\times \overline{b}\right)\\ =\overline{a}\left(3\overline{a}\times \overline{b}-0\right)\\ =3\overline{a}\times \overline{a}\times \overline{b}\\ =0\times \overline{b}\end{array}$

$=0$ R.H.S

Proved.