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)

Rate this question :

How useful is this solution?
We strive to provide quality solutions. Please rate us to serve you better.
Related Videos
Revision of straight lines important formulas in one shot59 mins
Concept Builder Questions of straight Lines (Quiz Session)55 mins
Magnetic field due to current carrying straight wire37 mins
Translation Process37 mins
Interactive Quiz on Bacterial Diseases29 mins
Immune System Disorders60 mins
Interactive Quiz on Cancer36 mins
Interactive Quiz on Magnetic field due to straight wire & circular loop41 mins
Interactive Quiz on Connective & Epithelial Tissue45 mins
Human health & diseases (Drug)56 mins
Try our Mini CourseMaster Important Topics in 7 DaysLearn from IITians, NITians, Doctors & Academic Experts
Dedicated counsellor for each student
24X7 Doubt Resolution
Daily Report Card
Detailed Performance Evaluation
view all courses