If $A = (k, 1, -1)$,$B = (2k, 0, 2)$,and $C = (2 + 2k, k, 1)$,and $AB \perp BC$,then the value of $k$ is:

Given three vectors $a=2 \hat{i}-\hat{j}+\hat{k}$,$b=\hat{i}+2 \hat{j}-\hat{k}$,and $c=\hat{i}+\hat{j}-2 \hat{k}$,a vector in the plane of $b$ and $c$ whose projection on $a$ is of magnitude $\sqrt{\frac{2}{3}}$ is

If $x$ and $y$ are two unit vectors and the angle between them is $\phi$,then $\frac{1}{2} |x - y| = $

If the resultant of two forces is of magnitude $P$ and equal to one of them and perpendicular to it,then the other force is

If $\theta$ is the angle between the vectors $\bar{a}$ and $\bar{b}$ where $|\bar{a}|=4, |\bar{b}|=3$ and $\theta \in \left(\frac{\pi}{4}, \frac{\pi}{3}\right)$,then $|(\bar{a}-\bar{b}) \times(\bar{a}+\bar{b})|^2+4(\bar{a} \cdot \bar{b})^2$ has the value

Let $\overrightarrow{a}=2 \hat{i}-\hat{j}+\hat{k}$,$\overrightarrow{b}=\hat{i}+2 \hat{j}-\hat{k}$ and $\overrightarrow{c}=\hat{i}+\hat{j}-2 \hat{k}$ be three vectors. $A$ vector in the plane of $\overrightarrow{b}$ and $\overrightarrow{c}$ whose projection on $\overrightarrow{a}$ is of magnitude $\sqrt{\frac{2}{3}}$,is