Three Dimension...

Direction cosines of $$3i$$ be

$$A(1, 0,0), B(0, 2, 0), C(0, 0, 3)$$ form the triangle $$ABC$$. Then the direction ratios of the line joining orthocenter and circumcentre of $$ABC$$ are

Assertion ($$A$$): 

If a ray makes angles $$\alpha, \beta, \gamma$$ and $$\delta$$ with the four diagonals of a cube and
$$\mathrm{A}:\cos^{2}\alpha+\cos^{2}\beta+\cos^{2}\gamma+\cos^{2}\delta$$
$$\mathrm{B}:\sin^{2}\alpha+\sin^{2}\beta+\sin^{2}\gamma+\sin^{2}\delta$$
$$\mathrm{C}:\cos 2\alpha+\cos 2\beta+\cos 2\gamma+\cos 2\delta$$
Arrange $$A,B,C$$ in descending order

Find the angle between the pair of lines $$\overrightarrow { r } =3i+2j-4k+\lambda \left( i+2j+2k \right) $$ and $$\overrightarrow { r } =5i-2k+\mu \left( 3i+2j+6k \right) $$.

A: The acute angle between the space diagonals (passing through opposite corners) of a cube.
B: The angle between the face diagonals of a cube.
C: The acute angle between a face diagonal and a space diagonal of a cube passing through the same corner of the cube.
Arrange A, B, C in ascending order

lf $$\alpha,\ \beta,\ \gamma$$ are the angles made by a line with the coordinate axes in the positive direction, then the range of $$\sin\alpha\sin\beta+\sin\beta\sin\gamma+\sin\gamma\sin\alpha$$ is

The intercepts made on the axes by the plane which bisects the line joining the points $$(1,2,3)$$ and $$(-3,4,5)$$ at right angles are

lf a line makes angles $$60^{o}, 45^{o}, 45^{o}$$ and $$\theta$$ with the four diagonals of a cube, then $$\sin^{2}\theta =$$

The direction ratios of the diagonal of a cube which joins the origin to the opposite corner are (when the three concurrent edges of the cube are coordinate axes)