Question

If $\overrightarrow{d}=5\overrightarrow{a}-3\overrightarrow{b}$, Find $\overrightarrow{d}\cdot (\overrightarrow{a}\times \overrightarrow{b})$

Answer 1

Given, $\overrightarrow{d}=5\overrightarrow{a}-3\overrightarrow{b}$.....91).

Now,

$\overrightarrow{d}.(\overrightarrow{a}\times \overrightarrow{b})$

$=5\overrightarrow{a}.(\overrightarrow{a}\times \overrightarrow{b})-3\overrightarrow{b}.(\overrightarrow{a}\times \overrightarrow{b})$ [ Using (1)]

$=5\left[\overrightarrow{a}\overrightarrow{a}\overrightarrow{b}\right]-3\left[\overrightarrow{b}\overrightarrow{a}\overrightarrow{b}\right]$ [ Using definition of vector tripple product]

$=5.0-3.0$ [ Since $\left[\overrightarrow{a}\overrightarrow{b}\overrightarrow{a}\right]=0$]

$=0$.