Three Dimension...

$$L_{1}$$ and $$L_{2}$$ are two lines whose vector equations are
$$L_{1} = \vec {r} = \lambda [(\cos \theta + \sqrt {3})\hat {i} + (\sqrt {2}\sin \theta)\hat {j} + (\cos \theta - \sqrt {3})\hat {k}]$$
$$L_{2} = \vec {r} = \mu (a\hat {i} + b\hat {j} + c\hat {k})$$, where $$\lambda$$ and $$\mu$$ are scalars and $$\alpha$$ is the acute angle between $$L_{1}$$ and $$L_{2}$$. If the angle $$'\alpha'$$ is independent of $$\theta$$ then the value of $$'\alpha'$$ is

The equation of the line parallel to $$\cfrac { x-3 }{ 1 } =\cfrac { y+3 }{ 5 } =\cfrac { 2z-5 }{ 3 } $$ and passing through the point $$(1,3,5)$$ in vector form, is:

If points $$P\left( 4,5,x \right) ,Q\left( 3,y,4 \right) $$ and $$ R\left( 5,8,0 \right) $$ are colinear, then the value of $$x+y$$ is

If $$\alpha,\beta,\gamma\in[0,2\pi]$$, then the sum of all possible values of $$\alpha, \beta,\gamma$$ if $$\sin \alpha=-\dfrac{1}{\sqrt{2}}$$, $$\cos \beta=-\dfrac{1}{2}$$, $$\tan \gamma=-\sqrt{3}$$, is

In $$\Delta ABC, |\bar{CB}| = a, |\bar{CA}| = b, |\bar{AB}| = c$$. $$CD$$ is median through the vertex $$C$$. Then $$\bar{CA}.\bar{CD}$$ equals.

The equation of plane passing through a point $$A(2, - 1, 3)$$ and parallel to the vectors $$a= (3, 0, - 1)$$ and $$b=(- 3, 2, 2)$$ is:

If a = 4i + 3j and b be two vectors perpendicular to each other on the xy- plane. Then, a vector in the same plane having projections 1 and 2 along a and b respectively, is 

The equation of the plane passing through (1, -2, 4), (3, -4, 5) and perpendicular to yz-plane is.

If a line makes angle $$90^{o},135^{o},45^{o}$$ with the $$X-$$,$$Y-$$ and $$Z-$$axes respectively, then its direction cosines are

In the isosceles $$\triangle$$ABC, $$|AB| = |BC|=8$$ and point E divides AB internally in the ratio 1 : 3 then the cosine of angel between CE and CA is (where, |CA| = 12)  ?