If $A=(-2,2,3), B=(3,2,2), C=(4,-3,5)$ and $D=(7,-5,-1)$,then the projection of $\overline{AB}$ on $\overline{CD}$ is

Let a vector $\overrightarrow{a}=\sqrt{2}\hat{i}-\hat{j}+\lambda\hat{k}$,$\lambda>0$,make an obtuse angle with the vector $\overrightarrow{b}=-\lambda^{2}\hat{i}+4\sqrt{2}\hat{j}+4\sqrt{2}\hat{k}$ and an angle $\theta$,$\frac{\pi}{6} < \theta < \frac{\pi}{2}$,with the positive $z$-axis. If the set of all possible values of $\lambda$ is $(\alpha, \beta)-\{\gamma\}$,then $\alpha+\beta+\gamma$ is equal to . . . . . . .

$OA$ and $OB$ are two vectors of magnitudes $5$ and $6$ respectively. If $\angle BOA = 60^{\circ}$,then $OA \cdot OB$ is equal to

The orthogonal projection of vector $\vec{a}$ on vector $\vec{b}$ is:

The vectors $(a \cdot b) c$ and $(a \cdot c) b$ are:

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