MYTUTOR SUBJECT ANSWERS

548 views

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 ...

1 year ago

Answered by George, an A Level Maths tutor with MyTutor

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

183 SUBJECT SPECIALISTS

£20 /hr

Oscar Z.

Degree: General Engineering (Bachelors) - Durham University

Subjects offered: Maths, Physics+ 1 more

Maths
Physics
Further Mathematics

“I am a second year General Engineering student at Durham University. I have a passion for science and technology and often find delight in finding out how things work. Richard Feynman (Nobel Laureate) puts it better! https://www.youtu...”

£24 /hr

Tom H.

Degree: Economics (Bachelors) - Durham University

Subjects offered: Maths, Spanish+ 3 more

Maths
Spanish
French
Economics
Business Studies

“Top tutor from Durham University, ready to help you improve your grades, all the way to A*.”

£24 /hr

Joe D.

Degree: Philosophy, Politics and Economics (Bachelors) - Durham University

Subjects offered: Maths, Physics+ 2 more

Maths
Physics
Chemistry
Biology

“About Me:I am currently studying Philosophy, Politics and Economics at Durham University. So although I now specialise in humanities, I have continued to study the sciences simultaneously and mydepth of experience will enable me to h...”

About the author

George R.

Currently unavailable: for regular students

Degree: Mathematics (Bachelors) - Cambridge University

Subjects offered: Maths, Physics+ 2 more

Maths
Physics
Further Mathematics
.STEP.

“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 I integrate by parts?

How would you solve (2x+16)/(x+6)(x+7) in partial fractions?

Find the derivative of f(x)=x^2log(2x)

How do I sketch the graph y = (x^2 + 4*x + 2)/(3*x + 1)

View A Level Maths tutors

Cookies:

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

mtw:mercury1:status:ok