00:01
Alright, so we're given the parametric equations of two lines.
00:06
We want to show that they're skew.
00:12
So we have to prove two things.
00:15
They're not intersecting and they're not parallel.
00:23
Okay, so first, not intersecting.
00:31
So to show they're not intersecting, it means that we can't find values of s and t.
00:39
So i just chose the name of the parameter for line 2 to be s so we don't get confused.
00:45
Because they're not going to have necessarily the same t.
00:49
So they should be different.
00:51
So to show they're not intersecting, i have to set the x, y, and z values equal to each other.
01:00
So i get 1 equals 1 plus s and t equals 2 minus s and 2 plus t equals minus 2s.
01:17
Okay, because if i can find values of s and t that satisfy all three equations, then the lines would intersect at that corresponding point.
01:28
Okay, so the first equation tells me s equals 0.
01:32
Second equation tells me t equals 2.
01:37
The third equation tells me that 4 equals 0.
01:41
We know that's ridiculous.
01:45
4 does not equal 0 and therefore they are not intersecting.
01:55
Okay, and then we also have to show they're not parallel.
02:03
Okay, so to show they're not parallel, the easiest thing to do is to write out the vector forms of the two equations of the two lines.
02:12
So for line 1, our vector form looks like 1, 0, 2 plus t times 0, 1, 1.
02:35
And then for line 2, our position vector is, we just read them right off, 1, 2, 0 plus s times 1 minus 1 minus 2.
03:03
Okay, so parallel means that these two vectors are proportional, which means that if i take one of them and multiply it by some number, i'll get the other one.
03:31
And that clearly is not ever going to happen, no matter what number i pick.
03:37
So they're not proportional and therefore they're not parallel.
03:50
And therefore, which is the answer we want, they are skew.
04:19
So in part b, we want to find the distance between them...