Angle between two Lines:
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Let q be angle between 2 straight lines AB and AC whose direction cosines are given whose direction cosines are l1, m1, n1 and l2, m2, n2 respectively, can be given by cosq = l1l2 + m1m2 + n1n2
If direction ratios of 2 lines are a1, b1, c1 and a2, b2, c2 are given, then angle between 2 lines can be given by
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Particular Results:
- We have, sin2θ = 1 - cos2θ
- Condition of perpendicularity:
If given lines are perpendicular, then q = 900 i.e. cosθ = 0
=> l1l2 + m1m2 + n1n2 = 0 or a1a2 + b1b2 + c1c2 = 0 .
- Condition of parallelism:
If given lines are parallel, then q = 00 i.e. sinθ = 0
=> (l1m2 - l2m1)2 + (m1n2 - m2n1)2 + (n1l2 - n2l1)2 = 0
which is true, only when
l1m2 - l2m1 = 0, m1n2 - m2n1 = 0 and n1l2 - n2l1 = 0 
Likewise, 
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Example: Show that 2 lines having direction ratios -1, 3, 2 and 2, 2, -2 are perpendicular.
Solution: a1a2 + b1b2 + c1c2 = (-1)(2) + (3)(2) + (2)(-2) = -2 + 6 - 4 = 0
∴lines are perpendicular.
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