$A$ force $\overrightarrow{F} = 2i + j - k$ acts at a point $A$,whose position vector is $2i - j$. The moment of $\overrightarrow{F}$ about the origin is:

If $\bar{a}=2 \hat{i}+3 \hat{j}+4 \hat{k}$,$\bar{b}=\hat{i}-2 \hat{j}-2 \hat{k}$,$\bar{c}=-\hat{i}+4 \hat{j}+3 \hat{k}$ and if $\bar{d}$ is a vector perpendicular to both $\bar{b}$ and $\bar{c}$,and $\bar{a} \cdot \bar{d}=18$,then $|\bar{a} \times \bar{d}|^2=$

Let $a = 2i + j - 2k$ and $b = i + j$. If $c$ is a vector such that $a \cdot c = |c|$,$|c - a| = 2\sqrt{2}$,and the angle between $(a \times b)$ and $c$ is $30^\circ$,then $|(a \times b) \times c| = $

Let $ABC$ be a triangle of area $15 \sqrt{2}$ and the vectors $\overrightarrow{AB}=\hat{i}+2 \hat{j}-7 \hat{k}$,$\overrightarrow{BC}=a \hat{i}+b \hat{j}+c \hat{k}$ and $\overrightarrow{AC}=6 \hat{i}+d \hat{j}-2 \hat{k}$,where $d>0$. Then the square of the length of the largest side of the triangle $ABC$ is:

If vectors $a, b,$ and $c$ represent the sides $BC, CA,$ and $AB$ of a triangle $ABC$ respectively,then which of the following is true?

If $a = (1, 1, 1)$ and $c = (0, 1, -1)$ are two vectors and $b$ is a vector such that $a \times b = c$ and $a \cdot b = 3$,then $b$ is equal to