Three Dimension...

The sum of the squares of sine of the angles made by the line $$AB$$ with $$OX, OY, OZ$$ where $$O$$ is the origin is

The direction ratios of a normal to the plane passing through $$(0,1,1), (1,1,2)$$ and $$(-1,2,-2)$$ are

The direction ratios of a normal to the plane through $$(1,\ 0,\ 0),\ (0,\ 1,\ 0)$$ which makes an angle of $$\displaystyle \dfrac{\pi}{4}$$ with the plane $$x+y=3$$ are

If $$l_{1},m_{1},n_{1}$$ and $$1_{2},m_{2},n_{2}$$ are the direction cosines of two lines, then $$(l_{1}l_{2}+m_{1}m_{2}+n_{1}n_{2})^{2}+\displaystyle\sum(m_{1}n_{2}-m_{2}n_{1})^{2}=$$

Find the direction cosines of vector $$\overrightarrow { r } $$ which is equally inclined to $$OX,OY$$ and $$OZ$$. Find total number of such vectors.

lf $$\theta $$ is the angle between two lines whose d.c.s are $$l_{1},m_{1},n_{1}$$ and $$l_{2}, m_{2}, n_{2}$$, then the d.cs of one of the angular bisectors of the two lines are

The product of the d.cs of the line which makes equal angles with $$ox, oy, oz$$ is

lf a line makes angles $$\displaystyle \dfrac{\pi}{12}, \displaystyle \dfrac{5\pi}{12}$$ with $$OY, OZ $$ respectively where $$O=({0}, 0,0)$$, then the angle made by that line with $$OX$$ is

The coordinates of a point P are $$(3,12,4)$$ w.r.t origin O, then the direction cosines of $$OP$$ are

lf a line makes angles $$\alpha, \beta,\gamma$$ with $$OX, OY, OZ$$ respectively where $${O}=(0,0,0)$$, then $$\cos 2\alpha+\cos 2 \beta+\cos 2 \gamma=$$