Question

find the direction cosines of the line which is equally inclined with co ordinate axes

Answer 1

If the line is equally inclined to all the coordinate axes, let it be inclined at α.

So the direction cosines are:
l = cos α
m = cos α
n = cos α

We know that l² + m² + n² = 1
⇒ cos² α + cos² α + cos² α = 1
⇒ 3cos² α = 1
⇒ cos² α = 1/3
⇒ cos α = $\pm \frac{1}{ \sqrt{3} }$

So the direction cosines are $( \frac{1}{ \sqrt{3}}, \frac{1}{ \sqrt{3}}, \frac{1}{ \sqrt{3} } )$ and $(- \frac{1}{ \sqrt{3}},- \frac{1}{ \sqrt{3}},- \frac{1}{ \sqrt{3}})$