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.

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