How do you show that two lines do, or do not intersect?

How you should approach a question of this type in an exam

Say you are given two lines Land Lwith equations

 r= (a1,b1,c1) + s(d1,e1,f1) and r= (a2,b2,c2) + t(d2,e2,f2) respectively and you are asked to deduce whether they do or do not intersect.

Then we need to either:

Find values of s and t such that both position vectors rand rare equal and thus giving us a point of intersection.

Or show that such a pair of s and t does not exist.

In both cases we try to find the s and t , we either succeed or we reach a contradiction - which shows that they cannot intersect.

What the best method is for doing this and how to display it to the examiner

For two lines to intersect, each of the three components of the two position vectors at the point of intersection must be equal.

Therefore we can set up 3 simultaneous equations, one for each component.

However we only have 2 unknowns to find (s and t) so we only need two of these equations, so we pick two of them. (Here I pick the first two)

We write:     a+ sd= a+ td2

                      b+ se= b+ te2

and we solve these in the usual way to find our s and t, showing our working.

Then we substitute in our values for s and t into our third equation:

If we have a point of intersection, then we should have one side equal to the other. We then may have to find the point of intersection, we do this by plugging our value of s into the equation of the first line, and t into the equation of the second. This gives us our point of intersection as they should be equal, if not , you have made a mistake in solving the two simultaneous equations. 

If the lines do not intersect, then both sides will not be equal and we will have something like '5=8'. We now write 'This is a contradiction, so the lines do not intersect.'

George R. A Level Further Mathematics  tutor, A Level Maths tutor, A ...

2 years ago

Answered by George, an A Level Maths tutor with MyTutor

Still stuck? Get one-to-one help from a personally interviewed subject specialist


£22 /hr

Samuel H.

Degree: Mathematics with International Study (Masters) - Exeter University

Subjects offered: Maths, Physics


“I have not only been able to teach AS Level students how to solve mechanics problems; I helped them understand the concepts behind the problems”

£20 /hr

Philip D.

Degree: Mathematics (Bachelors) - Exeter University

Subjects offered: Maths, Further Mathematics

Further Mathematics

“Hey, I'm Phil, a mathematics student at the University of Exeter. Unsurprisingly maths has always been a subject that has fascinated me given its intrinsic and logical nature, and I hope that I can help develop your understanding of t...”

£26 /hr

Imogen W.

Degree: Chemistry (Masters) - University College London University

Subjects offered: Maths, Chemistry


“I am very enthusiastic and patient. I want my students to be able to understand their studies so that they can enjoy them as much as I do.”

About the author

George R.

Currently unavailable: for regular students

Degree: Mathematics (Bachelors) - Cambridge University

Subjects offered: Maths, Physics+ 2 more

Further Mathematics

“A friendly and patient mathematics undergraduate, I enjoy helping students improve their understanding and gain confidence.”

You may also like...

Posts by George

Derive the formula for the maximum kinetic energy of an electron emitted from a metal with work function energy p , that is illuminated by light of frequency f.

How do you show that two lines do, or do not intersect?

Other A Level Maths questions

How do you 'rationalise the denominator'?

How do you differentiate (3x+cos(x))(2+4sin(3x))?

What is integration by parts?

When and how do I use the product rule for differentiation?

View A Level Maths tutors


We use cookies to improve our service. By continuing to use this website, we'll assume that you're OK with this. Dismiss