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: L1 and L2 with equations and you are asked to deduce whether or not they intersect.
For example:
L1: r1 = (a1,b1,c1) + s(d1,e1,f1)
L2: r2 = (a2,b2,c2) + t(d2,e2,f2)
Then we need to either:
- Find values of s and t such that both position vectors r1 and r2 are 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 and 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:
a1 + sd1 = a2 + td2
b1 + se1 = b2 + 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.'
