Show that the lines and intersect each other. Also, find the point of their intersection.
Given: The equations of the two lines are
To Prove: The two lines intersect and to find their point of intersection.
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 + 2, 4λ1 + 3)
A point on the second line is (5λ2 + 4, 2λ2 + 1, λ2)
If they intersect they should have a common point.
2λ1 + 1 = 5λ2 + 4 ⇒ 2λ1 – 5λ2 = 3 … (1)
3λ1 + 2 = 2λ2 + 1 ⇒ 3λ1 - 2λ2 = -1 … (2)
Solving (1) and (2),
-11λ2 = 11
λ2 = -1
Therefore, λ1 = -1
Substituting for the z coordinate, we get
4λ1 + 3 = -1 and λ2 = -1
So, the lines intersect and their point of intersection is (-1, -1, -1)
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