Let $\overrightarrow{a}=3\hat{i}-\hat{j}+2\hat{k}$,$\overrightarrow{b}=\overrightarrow{a}\times(\hat{i}-2\hat{k})$ and $\overrightarrow{c}=\overrightarrow{b}\times\hat{k}$. Then the projection of $\overrightarrow{c}-2\hat{j}$ on $\overrightarrow{a}$ is:

Let $\overline{a}=\hat{i}+2 \hat{j}-\hat{k}$ and $\overline{b}=\hat{i}+\hat{j}-\hat{k}$ be two vectors. If $\overline{c}$ is a vector such that $\overline{b} \times \overline{c}=\overline{b} \times \overline{a}$ and $\overline{c} \cdot \overline{a}=0$,then $\overline{c} \cdot \overline{b}$ is

Let $\vec{a}$ and $\vec{b}$ be the vectors along the diagonals of a parallelogram having area $2 \sqrt{2}$. Let the angle between $\vec{a}$ and $\vec{b}$ be acute. Given $|\vec{a}|=1$ and $|\vec{a} \cdot \vec{b}|=|\vec{a} \times \vec{b}|$. If $\vec{c}=2 \sqrt{2}(\vec{a} \times \vec{b})-2 \vec{b}$,then find the angle between $\vec{b}$ and $\vec{c}$.

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 the vectors $\overline{a}, \overline{b}, \overline{c}$ and $\overline{d}$ be such that $(\overline{a} \times \overline{b}) \times(\overline{c} \times \overline{d})=\overline{0}$. Let $P_1$ and $P_2$ be the planes determined by the pair of vectors $\overline{a}, \overline{b}$ and $\overline{c}, \overline{d}$ respectively,then the angle between $P_1$ and $P_2$ is

If $a, b$ and $c$ are position vectors of the vertices of $\triangle ABC$,then $\frac{(a-c) \times (b-a)}{(b-a) \cdot (c-a)} = $