Question

For the vectors $\overline{x}$ and $\overline{y}$, $[2\overline{x}+3\overline{y}\overline{x}\overline{y}]=$ _____.

• A. $0$
• B. $6(\overline{x}\times \overline{y})$
• C. $6{\left|\overline{x}\right|}^2$
• D. $1$

Answer 1

The correct option is A $0$

$[2\overline{x}+3\overline{y}\overline{x}\overline{y}]=(2\overline{x}+3\overline{y}).(\overline{x}\times \overline{y})=2\overline{x}.(\overline{x}\times \overline{y})+3\overline{y}(\overline{x}\times \overline{y})$
$=2\left[\overline{x}\overline{x}\overline{y}\right]+3\left[\overline{y}\overline{x}\overline{y}\right]=2\left(0\right)+3\left(0\right)=0$
We have $\forall \alpha ,\beta \in R$ then $[\alpha \overline{x}+\beta \overline{y}\overline{x}\overline{y}]=0$
Here take $\alpha =2$ and $\beta =3$
$[2\overline{x}+3\overline{y}\overline{x}\overline{y}]=0$
$=0$