Parallel Lines:
As the parallel lines have the same direction, it follows those direction cosines of 2 or more parallel straight lines are the same. In the case of lines, which do not pass through the origin, we draw a parallel line passing through origin and direction cosines of that line can be found.
Example: Find the direction cosines of 2 lines that are connected by the relations l-5m + 3n = 0 and 7l2 + 5m2 - 3n2 = 0.
Solution : The given relations are
l-5m + 3n = 0 => l = 5m - 3n ......(1)
and 7l2 + 5m2 - 3n2 = 0 ......(2)
By putting the value of l from (1) in (2), we get
7(5m - 3n)2 + 5m2 - 3n2 = 0
or, 180m2 - 210mn + 60n2 = 0 or, (2m - n)(3m - 2n) = 0
∴ m/n = 1/2 or 2/3
when m/n = 1/2 that is n = 2m
∴ l = 5m - 3n = -m or 1/m = -1
thus m/n = 1/2 and 1/m = -1 giving l/-1 = m/1 = n/2
or, 
So, direction cosines of 1 line are 
Again when m/n = 1/2 or n = 3m/2
Direction Cosine of a Line joining two given Points:
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The direction ratios of the line PQ joining P (x1, y1, z1) and Q(x2, y2, z2) are x2 - x1 = a(say), y2 - y1 = b (say) and
z2 - z1 = c (say).
Then the direction cosines are
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