00:01
In this video, we are stationed at the point 450, and we want to look at the point negative 2, negative 3, 2, but there's a ball of radius 1 at the origin that we can't see through.
00:16
So we want to know if our vision is blocked by this ball.
00:19
So what i'm going to do is i'm going to write the equation, the parametric equations for the line connecting these two.
00:27
And so that's just given by you start with your point 450, and then you just add t with.
00:36
Which is the time times the vector that points from 450 to negative 2, negative 3, negative 2.
00:45
So that's negative 2 minus 4, negative 3 minus 5, 2 minus 0.
00:58
So we get 4 -50 plus t times negative 6, negative 8, 2.
01:08
So we can write this as x equals 4 minus 6 t, y is equal to 5 minus 8 t, and z is equal to 2 t.
01:28
And so the question is, does the line intersect with the ball? so basically the question is, is for any value of t, is x, y, z within this ball of radius 1 centered? at the origin.
02:00
So the way we can do that is we can take the distance of any point on our line to the origin using the pythagorean theorem.
02:08
So we'll call it distance, and we can actually just square it so we don't have to deal with the square root.
02:15
And using the pythagorean theorem, that's just 4 minus 6t squared plus 5 minus 8t squared plus 2t squared.
02:30
So this is the distance, and the question is, and i'll use this as a question mark, is this ever less than or equal to 1? in which case that means it's within the ball of radius 1.
02:43
So i won't go through the details.
02:45
You can minimize this.
02:46
It's just a quadratic function after you do all of the algebra.
02:50
But if you do that, you'll see that its minimum value is about 1 .6, and therefore it does not intersect.
03:09
It is visible...