If $|\bar{u}| = 8$ and $|\bar{v}| = 12$ with an angle of $150^{\circ}$ between them,then find $|\bar{u} \times \bar{v}|$.

Let $O$ be the origin,and $\overline{OX}, \overline{OY}, \overline{OZ}$ be three unit vectors in the directions of the sides $QR, RP, PQ$,respectively,of a triangle $PQR$.
$(1)$ Find $|\overline{OX} \times \overline{OY}|$.
$[A] \sin(P+Q)$
$[B] \sin 2R$
$[C] \sin(P+R)$
$[D] \sin(Q+R)$
$(2)$ If the triangle $PQR$ varies,then find the minimum value of $\cos(P+Q) + \cos(Q+R) + \cos(R+P)$.
$[A] -\frac{5}{3}$
$[B] -\frac{3}{2}$
$[C] \frac{3}{2}$
$[D] \frac{5}{3}$
Select the correct options for $(1)$ and $(2)$.

Let $a$,$b$,and $c$ be real numbers such that $a^2 + b^2 + c^2 = 1$. Then the maximum value of $(4b - 3c)^2 + (4a - 2c)^2 + (3a - 2b)^2$ is:

If the vertices of $\Delta ABC$ are $A(1, -1, 2)$,$B(2, 0, -1)$,and $C(0, 2, 1)$,then what is the area of the triangle?

If $\overrightarrow{u}=\overrightarrow{a}-\overrightarrow{b}$,$\overrightarrow{v}=\overrightarrow{a}+\overrightarrow{b}$,$|\overrightarrow{a}|=|\overrightarrow{b}|=2$,then $|\overrightarrow{u} \times \overrightarrow{v}|$ is equal to

Let $\vec{a} = 4\hat{i} - \hat{j} + 3\hat{k}$, $\vec{b} = 10\hat{i} + 2\hat{j} - \hat{k}$ and a vector $\vec{c}$ be such that $2(\vec{a} \times \vec{c}) + 3(\vec{b} \times \vec{c}) = \vec{0}$. If $\vec{a} \cdot \vec{c} = 15$, then $\vec{c} \cdot (\hat{i} + \hat{j} - 3\hat{k})$ is equal to: