Let three vectors $\vec{a}, \vec{b}$ and $\vec{c}$ be such that $\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{b}}=\overrightarrow{\mathrm{c}}, \overrightarrow{\mathrm{b}} \times \overrightarrow{\mathrm{c}}=\overrightarrow{\mathrm{a}}$ and $|\overrightarrow{\mathrm{a}}|=2$. Then which one of the following is not true ?
Correct Option: , 4
(1) $\overrightarrow{\mathrm{a}} \times((\overrightarrow{\mathrm{b}}+\overrightarrow{\mathrm{c}}) \times(\overrightarrow{\mathrm{b}}-\overrightarrow{\mathrm{c}}))$
$=\overrightarrow{\mathrm{a}}(-\overrightarrow{\mathrm{b}} \times \overrightarrow{\mathrm{c}}+\overrightarrow{\mathrm{c}} \times \overrightarrow{\mathrm{b}})=-2(\overrightarrow{\mathrm{a}} \times(\overrightarrow{\mathrm{b}} \times \overrightarrow{\mathrm{c}}))$
$=-2(\vec{a} \times \vec{a})=\overrightarrow{0}$
(2) Projection of $\vec{a}$ on $\vec{b} \times \vec{c}$
$=\frac{\overrightarrow{\mathrm{a}} \cdot(\overrightarrow{\mathrm{b}} \times \overrightarrow{\mathrm{c}})}{|\overrightarrow{\mathrm{b}} \times \overrightarrow{\mathrm{c}}|}=\frac{\overrightarrow{\mathrm{a}} \cdot \overrightarrow{\mathrm{a}}}{|\overrightarrow{\mathrm{a}}|}=|\overrightarrow{\mathrm{a}}|=2$
(3) $[\vec{a} \vec{b} \vec{c}]+[\vec{c} \vec{a} \vec{b}]=2[\vec{a} \vec{b} \vec{c}]=2 \vec{a} \cdot(\vec{b} \times \vec{c})$
$=2 \overrightarrow{\mathrm{a}} \cdot \overrightarrow{\mathrm{a}}=2|\overrightarrow{\mathrm{a}}|^{2}=8$
(4) $\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{b}}=\overrightarrow{\mathrm{c}}$ and $\overrightarrow{\mathrm{b}} \times \overrightarrow{\mathrm{c}}=\overrightarrow{\mathrm{a}}$
$\Rightarrow \vec{a}, \vec{b}, \vec{c}$ are mutually $\perp$ vectors.
$\therefore|\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{b}}|=|\overrightarrow{\mathrm{c}}| \Rightarrow|\overrightarrow{\mathrm{a}}||\overrightarrow{\mathrm{b}}|=|\overrightarrow{\mathrm{c}}| \Rightarrow|\overrightarrow{\mathrm{b}}|=|\overrightarrow{\mathrm{c}}| / 2$
Also, $|\overrightarrow{\mathrm{b}} \times \overrightarrow{\mathrm{c}}|=|\overrightarrow{\mathrm{a}}| \Rightarrow|\overrightarrow{\mathrm{b}}||\overrightarrow{\mathrm{c}}|=2 \Rightarrow|\overrightarrow{\mathrm{c}}|=2 \&|\overrightarrow{\mathrm{b}}|=1$
$|3 \vec{a}+\vec{b}-2 \vec{c}|^{2}=(3 \vec{a}+\vec{b}-2 \vec{c}) \cdot(3 \vec{a}+\vec{b}-2 \vec{c})$
$=9|\overrightarrow{\mathrm{a}}|^{2}+|\overrightarrow{\mathrm{b}}|^{2}+4|\overrightarrow{\mathrm{c}}|^{2}$
$=(9 \times 4)+1+(4 \times 4)$
$=36+1+16=53$