Let the vectors $a, b, c$ and $d$ be such that $(a \times b) \times (c \times d) = 0$. Let $P_1$ and $P_2$ be planes determined by the pairs of vectors $(a, b)$ and $(c, d)$ respectively. Then the angle between $P_1$ and $P_2$ is:

The points $A(a), B(b), C(c)$ will be collinear if

Let $\vec{a}=2\hat{i}-5\hat{j}+5\hat{k}$ and $\vec{b}=\hat{i}-\hat{j}+3\hat{k}$. If $\vec{c}$ is a vector such that $2(\vec{a}\times\vec{c})+3(\vec{b}\times\vec{c})=\vec{0}$ and $(\vec{a}-\vec{b})\cdot\vec{c}=-97$,then $|\vec{c}\times \hat{k}|^{2}$ is equal to

The area of the triangle having vertices as $i - 2j + 3k,$ $- 2i + 3j - k,$ and $4i - 7j + 7k$ is

Let $\overrightarrow{OA}=2 \overrightarrow{a}$,$\overrightarrow{OB}=6 \overrightarrow{a}+5 \overrightarrow{b}$ and $\overrightarrow{OC}=3 \overrightarrow{b}$,where $O$ is the origin. If the area of the parallelogram with adjacent sides $\overrightarrow{OA}$ and $\overrightarrow{OC}$ is $15$ sq. units,then the area (in sq. units) of the quadrilateral $OABC$ is equal to :

If $\overline{a}=3 \hat{i}-5 \hat{j}$ and $\overline{b}=6 \hat{i}-3 \hat{j}$ are two vectors and $\overline{c}$ is a vector such that $\overline{c}=\overline{a} \times \overline{b}$,then the ratio $a: b: c$ is: