If $i, j, k$ are unit orthonormal vectors and $a$ is a vector,if $a \times r = j$,then $a \cdot r$ is

If $3$ vectors $a, b, c$ are such that $a \neq 0$ and $a \times b = 2(a \times c)$,$|a| = 1$,$|c| = 1$,$|b| = 4$,and the angle between $b$ and $c$ is $\cos^{-1}\left(\frac{1}{4}\right)$,and $b - 2c = \lambda a$,then $\lambda = $

The work done in moving an object along the vector $\vec{d} = 3i + 2j - 5k$,if the applied force is $\vec{F} = 2i - j - k$,is

The position vectors of points $A, B, C$ are given by $2\hat{i} - \hat{j} + \hat{k}$,$\hat{i} - 3\hat{j} - 5\hat{k}$,and $a\hat{i} - 3\hat{j} + \hat{k}$ respectively. If these points form a right-angled triangle with $\angle C = \pi/2$,find the value of $a$.

If three consecutive vertices of a parallelogram are $A(4,3,5)$,$B(0,6,0)$,$C(-8,1,4)$ and $D$ is the fourth vertex,then the angle between $AC$ and $BD$ is

$M$ and $N$ are the midpoints of the sides $BC$ and $CD$ of a parallelogram $ABCD$ respectively,then $\overline{AM} + \overline{AN} =$