Q. 13

# Show that the lines and intersect each other. Also, find the point of their intersection.

Given: The equations of the two lines are and To Prove: The two lines intersect and to find their point of intersection.

Formula Used: Equation of a line is

Vector form: Cartesian form: where is a point on the line and b1 : b2 : b3 is the direction ratios of the line.

Proof:

Let  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.

1 + 1 = 5λ2 + 4 1 – 5λ2 = 3 … (1)

1 + 2 = 2λ2 + 1 1 - 2λ2 = -1 … (2)

Solving (1) and (2),

-11λ2 = 11

λ2 = -1

Therefore, λ1 = -1

Substituting for the z coordinate, we get

1 + 3 = -1 and λ2 = -1

So, the lines intersect and their point of intersection is (-1, -1, -1)

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