If $a, b, c$ and $d$ are vectors in which $|d|=1$ and given $a+b+c=s d$ and $b+c+d=a$,with $a \cdot d=4$,then $s$ is equal to

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

Let $\bar{a} = a_1 \hat{i} + a_2 \hat{j} + a_3 \hat{k}$,where $a_1, a_2, a_3$ and $|\bar{a}|$ are rational numbers. If $\bar{a}$ makes an angle of $45^{\circ}$ with $\bar{b} = \sqrt{2} \hat{i} + 3 \sqrt{2} \hat{j} + 4 \hat{k}$,then $\bar{a}$ lies in:

The vector $2i + j - k$ is perpendicular to $i - 4j + \lambda k,$ if $\lambda = $

The angle between two vectors $\vec{a} = \hat{i} - \hat{j} + \hat{k}$ and $\vec{b} = \hat{i} + \hat{j} - \hat{k}$ is . . . . . . .

Let $b = 4i + 3j$ and $c$ be two vectors perpendicular to each other in the $xy$-plane. All vectors in the same plane having projections $1$ and $2$ along $b$ and $c$ respectively,are given by