Given: The equations of the two lines are
To Prove: the lines do not intersect each other.
Formula Used: Equation of a line is
where is a point on the line and b1 : b2 : b3 is the direction ratios of the line.
So a point on the first line is (2λ1 + 1, 3λ1 – 1, λ1)
A point on the second line is (5λ2 - 1, λ2 + 1, 2)
If they intersect they should have a common point.
2λ1 + 1 = 5λ2 - 1 ⇒ 2λ1 – 5λ2 = -2 … (1)
3λ1 – 1 = λ2 + 1 ⇒ 3λ1 - λ2 = 2 … (2)
Solving (1) and (2),
-13λ2 = -10
Substituting for the z coordinate, we get
and z = 2
So, the lines do not intersect.
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