Three Dimension...

A plane passing through $$(-1, 2, 3)$$ and whose normal makes equal angle with the coordinate axes is

The direction cosine of a line which is perpendicular to both the lines whose direction ratios are $$1, 2, 2$$ and $$0, 2, 1$$ are

Three district points A, B and C with p.v.s. and $$\displaystyle \vec { a } ,\vec { b } $$ and $$\displaystyle \vec { c } $$ respectively are collinear if there exist non-zero scalars x, y, z such that

The direction ratios of two lines AB, AC are 1, -1, -1 and 2, -1, 1. The direction ratios of the normal to the plane ABC are

If $$A(3, 4, 5), B(4, 6, 3), C(-1, 2, 4)$$ and $$D(1, 0, 5)$$ are such that the angle between the lines $$\overline{DC}$$ and $$\overline{AB}$$ is $$\theta$$ then $$cos\,\theta =$$

If a line makes angles $$\alpha, \beta, \gamma$$ with the coordinate axes, then the value of $$\cos 2\alpha + \cos 2\beta + \cos 2\gamma$$ is

The angle between two diagonals of a cube is.

If $$\cos { \alpha  } ,\cos { \beta  } ,\cos { \gamma  } $$ are the direction cosines of a vector $$\vec { a } $$, then $$\cos { 2\alpha  } +\cos { 2\beta  } +\cos { 2\gamma  } $$ is equal to

The vector equation of the plane which is at a distance of $$\cfrac { 3 }{ \sqrt { 14 }  } $$ from the origin and the normal from the origin is $$2\hat { i } -3\hat { j } +\hat { k } $$ is

If the angles made by a straight line with the coordinate axes are $$\alpha, \dfrac{\pi}{2}-\alpha, \beta$$ then $$\beta=$$