Question

The number of lines which are equally inclined to all the coordinate axes

Answer 1

Hello,
If a line makes angles α,β,ɣ with the axes we have α = β = ɣ .
cos α = cos β = cos ɣ or l = m = n .
l² + m² + n² = 1
l² + l² + l² = 1
3l² = 1
l² = 1/3
l = ± 1/ √3

The directional cosine's of lines are ( ± 1/ √3, ± 1/ √3, ± 1/ √3)

To find such lines are we have following combinations of signs l, m, n are possible
( +++ , ++-, + - -,- - - , - + + , - - +, + - + , - + + ) so total 8

But note that( l, m, n) and (-l , -m , -n ) represent the directionla.cosine's of the same line on the 2 sides of coordinate axis.
Since there are four different groups of signs, so there can be four different lines which makes equal angles with axes ANs = 4