Let `l_(1),m_(1),n_(1),l_(2),m_(2),n_(2)" and "l_(3),m_(3),n_(3)` be the durection cosines of three mutually perpendicular lines. Show that the direction ratios of the line which makes equal angles with each of them are
`(l_(1)+l_(2)+l_(3)),(m_(1)+m_(2)+m_(3)),(n_(1)+n_(2)+n_(3))`.

If l_(1), m_(1), n_(1) and l_(2), m_(2), n_(2) are direction cosines of two mutually perpendicular straight lines, then prove that the direction cosines of the straight line which is perpendicular to both the given lines are pm (m_(1)n_(2)-m_(2)n_(1)), pm (n_(1)l_(2)-n_(2)l_(1)), pm (l_(1)m_(2)-l_(2)m_(1))

Let l_(1), m_(1), n_(1) and l_(2), m_(2), n_(2) be the direction ratios of two given straight lines. Find the direction ratios of a straight line which is perpendicular to both the given straight lines.

Find the direction cosines of the two lines which are connected by the relations l-5m+3n=0 and 7l^(2)+5m^(2)-3n^(2)=0-

Show that the pair of lines whose direction cosines are given by 3lm-4ln+mn=0 and l+2m+3n=0 are perpendicular to each other.

O is the origin and lines OA, OB and OC have direction cosines l_(r),m_(r)andn_(r)(r=1,2 and3) . If lines OA', OB' and OC' bisect angles BOC, COA and AOB, respectively, prove that planes AOA', BOB' and COC' pass through the line (x)/(l_(1)+l_(2)+l_(3))=(y)/(m_(1)+m_(2)+m_(3))=(z)/(n_(1)+n_(2)+n_(3)) .

Show that the pair of lines whose direction cosines are given by 3 lm-4ln+mn =0 and l+2m+3n = 0 are perpendicular to each other.

The angle between the lines whose direction cosines satisfy the equations l + m + n = 0 and l^(2) = m^(2) + n^(2) is

In the adjacent figure, a line n falls on lines l and m such that the sum of the interior angles 1 and 2 is less than 180°, then what can you say about lines l and m.

The straight lines (x-x_(1))/(l_(1))=(y-y_(1))/(m_(1))=(z-z_(1))/(n_(1))" and "(x)/(l_(2))=(y)/(m_(2))=(z)/(n_(2)) will be perpendicular, if -

An electron is in one of the 3d orbitals. Give the possible values on n, l and m_(l) for this electron.