How do I show two lines are skew?

Skew lines in 3 dimensions are those which are not parallel and do not intersect.

Let's take two example lines: l1 = (2, 2, -1) + λ(3, 1, -3) and l2 = (1, 0, 1) + µ(6, -4, 9)

First we need to show that they are not parallel. To do this we take the direction vectors (the second part with λ or µ constats) and check that one is not a multiple of the other. You can probably see by inspection that this is the case here. To be sure you can divide the first component of one by the first of the other and then check that this is not the same for both of the other components. 

3/6 = 1/2, 1/-4 = -1/4 and -3/9 = -1/3. Here dividing the components by eachother shows that one direction vector is not a multiple of the other since the values are not all the same. So l1 and l2 are not parallel.

Next we need to show that they don't intersect. To do this we can set up three simultaneous equations. Equating the x component of one line to the other and the same for y and z . For example with l1 and l2:

2 + 3λ = 1 + 6µ, equation 1,
2 + λ = 0 - 4µ, equation 2,
-1 - 3λ = 1 + 9µ, equation 3.

To show these don't intersect we need to show that these three equations aren't consistent (so the lines can't cross). One simple way to do this is to determine the value of λ and µ using two of the three equations, then substitute these values into the third equation you haven't used. If the equation is incorrect then they don't intersect. 

For example here I used equations 1 and which gave µ = -1 and λ = -7/3 (I did this by subtracting 1 from 3). If we substitute these values into equation we get 2*(-7/3) = -4*(-1) which gives -1/3 = 4, which is ofcourse incorrect and so these equations aren't consistent and the lines don't intersect. Therefore in this example l1 and l2 are skew.

DM
Answered by David M. Maths tutor

116938 Views

See similar Maths A Level tutors

Related Maths A Level answers

All answers ▸

Write tan(3x) in terms of tan(x). Hence show that the roots of t^3 - 3t^2 - 3t + 1 = 0 are tan(pi/12), tan(5pi/12) and tan(3pi/4)


AQA PC4 2015 Q5 // A) Find the gradient at P. B) Find the equation of the normal to the curve at P C)The normal P intersects at the curve again at the point Q(cos2q, sin q) Hence find the x-coordinate of Q.


A child of m1=48 kg, is initially standing at rest on a skateboard. The child jumps off the skateboard moving horizontally with a speed v1=1.2 ms^-1. The skateboard moves with a speed v2=16 ms^-1 in the opposite direction. Find the mass of the skateboard.


The curve C has a equation y=(2x-3)^5; point P (0.5,-32)lies on that curve. Work out the equation to the tangent to C at point P in the form of y=mx+c


We're here to help

contact us iconContact ustelephone icon+44 (0) 203 773 6020
Facebook logoInstagram logoLinkedIn logo

MyTutor is part of the IXL family of brands:

© 2025 by IXL Learning