Angle Between Two Lines

Three Dimensional Geometry of Class 12

Angle Between Two Lines 

Vector form

Let the vector equations of the two lines be Angle between two lines and Angle between two lines . These two lines are parallel to the vectors Angle between two lines and Angle between two lines respectively. Therefore, angle between these two lines is equal to the angle between Angle between two lines and Angle between two lines. Thus, if θ is the angle between the given lines, then

cosθ = Angle between two lines

In Cartesian Form it is given as

cos θ = l1l2 + m1m2 + n1n2

= Angle between two lines, where l1, m1, n1 & l2, m2, n2 are direction cosines and a1, b1, c1 & a2, b2, c2 are direction ratios of given lines.

If the lines are perpendicular, then Angle between two lines.Angle between two lines = 0. (vector form)

l1l2 + m1m2 + n1n2 = 0 or a1a2 + b1b2 + c1c2 = 0 (cartesian form)

If the lines are parallel, then Angle between two lines and Angle between two lines are parallel, therefore Angle between two lines = λAngle between two lines for some

scalar λ. i.e., Angle between two lines

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