If the angle between the lines, $${x \over 2} = {y \over 2} = {z \over 1}$$

and $${{5 - x} \over { - 2}} = {{7y - 14} \over p} = {{z - 3} \over 4}\,\,$$ is $${\cos ^{ - 1}}\left( {{2 \over 3}} \right),$$ then p is equal to :

The length of the projection of the line segment joining the points (5, -1, 4) and (4, -1, 3) on the plane, x + y + z = 7 is :

If L₁ is the line of intersection of the planes 2x - 2y + 3z - 2 = 0, x - y + z + 1 = 0 and L₂ is the line of intersection of the planes x + 2y - z - 3 = 0, 3x - y + 2z - 1 = 0, then the distance of the origin from the plane, containing the lines L₁ and L₂, is :

An angle between the lines whose direction cosines are gien by the equations,
$$l$$ + 3m + 5n = 0 and 5$$l$$m $$-$$ 2mn + 6n$$l$$ = 0, is :