If the vectors $\hat{i}-2x\hat{j}-3y\hat{k}$ and $\hat{i}+3x\hat{j}+2y\hat{k}$ are orthogonal to each other,then the locus of the point $(x, y)$ is

If $P(3, 4, 5)$,$Q(4, 6, 3)$,$R(-1, 2, 4)$,and $S(1, 0, 5)$,then the projection of the vector $\vec{RS}$ on the vector $\vec{PQ}$ is:

If $a = i - j$ and $b = i + k$,then a unit vector coplanar with $a$ and $b$ and perpendicular to $a$ is

The angle between the vectors $3\,i + j + 2\,k$ and $2\,i - 2\,j + 4\,k$ is

If in a right-angled triangle $ABC$,the hypotenuse $|\overrightarrow{AB}| = p$,then $\overrightarrow{AB} \cdot \overrightarrow{AC} + \overrightarrow{BC} \cdot \overrightarrow{BA} + \overrightarrow{CA} \cdot \overrightarrow{CB} = $

If $a(\vec{\alpha} \times \vec{\beta}) + b(\vec{\beta} \times \vec{\gamma}) + c(\vec{\gamma} \times \vec{\alpha}) = \overrightarrow{0}$, where $a, b, c$ are non-zero scalars, then the vectors $\vec{\alpha}, \vec{\beta}, \vec{\gamma}$ are