If $a = i + j + k$,$a \cdot b = 1$ and $a \times b = j - k$,then $b = $

If $\overline{a}=2 \hat{i}+3 \hat{j}+4 \hat{k}$,$\overline{b}=\hat{i}-2 \hat{j}+\hat{k}$,and $\overline{c}=\hat{i}+\hat{j}-\hat{k}$ are three vectors,and there exists a vector $\overline{r}$ such that $\overline{r} \times \overline{a}=\overline{b}$ and $\overline{r} \cdot \overline{c}=3$,then the value of $|\overline{r}|$ is:

If a particle is acted upon by forces of magnitude $6$ and $7$ units in the directions of $-\hat{i} - 2\hat{j} + 2\hat{k}$ and $2\hat{i} - 3\hat{j} - 6\hat{k}$ respectively,and it undergoes a displacement from point $P(2, -1, -3)$ to $Q(5, -1, 1)$,then the work done by the forces is .......... units.

Let $\bar{a} = 4\bar{i} + 3\bar{j}$ and $\bar{b}$ be two perpendicular vectors in the $XOY$-plane. $A$ vector $\bar{c}$ in the same plane and having projections $1$ and $2$ respectively on $\bar{a}$ and $\bar{b}$ is

Given three vectors $\bar{a}, \bar{b}, \bar{c}$,two of which are collinear. If $\bar{a}+\bar{b}$ is collinear with $\bar{c}$ and $\bar{b}+\bar{c}$ is collinear with $\bar{a}$,and $|\bar{a}|=|\bar{b}|=|\bar{c}|=\sqrt{2}$,then $\bar{a} \cdot \bar{b}+\bar{b} \cdot \bar{c}+\bar{c} \cdot \bar{a}=$

$\vec{a}, \vec{b}, \vec{c}$ are three unit vectors such that $|\vec{a}+\vec{b}+\vec{c}|=1$ and $\vec{a}$ is perpendicular to $\vec{b}$. If $\vec{c}$ makes angles $\alpha, \beta$ with $\vec{a}, \vec{b}$ respectively, then $\cos \alpha+\cos \beta=$