29-30 Show that the lines L1 and L2 intersect, and find...

29-30 Show that the lines $L_1$ and $L_2$ intersect, and find their point of intersection. 29. $L_1$: $x = 2 + t$, $y = 2 + 3t$, $z = 3 + t$ $L_2$: $x = 2 + t$, $y = 3 + 4t$, $z = 4 + 2t$ 30. $L_1$: $(x, y, z) = (-5, 7, 3) + t(4, -3, 1)$ $L_2$: $(x, y, z) = \left( -\frac{19}{4}, -\frac{3}{2}, 0 \right) + t(1, 4, 2)$

Transcript

00:02 Hello, we are given two lines in r3.

00:05 These are lines in three -dimensional space and we have to show that these two lines intersect first and we have to find a point of intersection.

00:17 The point we should recall is that two lines in the plane, unless they are parallel, they have to intersect.

00:26 But in the three -dimensional space, two lines which are not parallel may not intersect.

00:32 We can have skew lines.

00:36 So, yeah, so intersection in three -dimensional space is much more non -intuitive.

00:43 So, let us find out, so we should keep this in mind.

00:46 So, let us find out if they intersect or not.

00:48 So, if these two things intersect, we should be able to, now there is one, before we go ahead, we have to make one a.

00:57 Instead of, we don't want to use the same parameter for both these equations because we are going to simultaneously solve them.

01:05 So, let us, for l2, let us use the parameter s.

01:09 So, let us rewrite the equations for l2 as x equal to 2 plus s, y equal to 3 plus 4s and z equal to 4 plus 2s.

01:22 Okay, we are going to use this equation of l2.

01:26 Now, when at the point of intersection the x coordinates will be equal, y coordinates will be equal and z coordinates will be equal.

01:34 That is, there will exist a t and there will exist a s such that when we plug in t into these equations and when we plug in s into these equations, they will give the same value of xyz.

01:49 So, let us see.

01:50 So, we are solving this.

01:52 Okay, so what we will have is that at the point of intersection, 2 plus t will be equal to 2 plus s and 2 plus 3t will be equal to 3 plus 4s and we will have that 3 plus t will be equal to 4 plus 2s.

02:13 So, we are solving these three equations simultaneously for t and s...

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