# Show that the lines and do not intersect each other.

Given: The equations of the two lines are

and

To Prove: the lines do not intersect each other.

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 – 1, λ1)

A point on the second line is (5λ2 - 1, λ2 + 1, 2)

If they intersect they should have a common point.

1 + 1 = 5λ2 - 1 1 – 5λ2 = -2 … (1)

1 – 1 = λ2 + 1 1 - λ2 = 2 … (2)

Solving (1) and (2),

-13λ2 = -10

Therefore,

Substituting for the z coordinate, we get

and z = 2

So, the lines do not intersect.

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