Let three vectors

Solution:

(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, $|\overrightarrow{\mathrm{b}} \times \overrightarrow{\mathrm{c}}|=|\overrightarrow{\mathrm{a}}| \Rightarrow|\overrightarrow{\mathrm{b}}||\overrightarrow{\mathrm{c}}|=2 \Rightarrow|\overrightarrow{\mathrm{c}}|=2 \&|\overrightarrow{\mathrm{b}}|=1$

$|3 \vec{a}+\vec{b}-2 \vec{c}|^{2}=(3 \vec{a}+\vec{b}-2 \vec{c}) \cdot(3 \vec{a}+\vec{b}-2 \vec{c})$

$=9|\overrightarrow{\mathrm{a}}|^{2}+|\overrightarrow{\mathrm{b}}|^{2}+4|\overrightarrow{\mathrm{c}}|^{2}$

$=(9 \times 4)+1+(4 \times 4)$

$=36+1+16=53$