Find an equation of the set of all points equidistant from th points A(-1,5,3) and B(6,2,-2) . D

Find an equation of the set of all points equidistant from...

Key Concepts

Distance Formula in Three Dimensions

This formula calculates the distance between any two points in space and is fundamental in problems involving equidistance. It provides a way to quantitatively compare distances from a variable point to fixed points by using the square root of the sum of squared differences in each coordinate direction.

Equidistant Locus

The equidistant locus is the set of all points that are the same distance from two fixed points. In three dimensions, this set typically forms a plane when dealing with non-collinear scenarios, as it captures all points that balance the distances to the two given points.

Perpendicular Bisector Plane

For two points in space, the perpendicular bisector plane is the geometric locus of all points that are equidistant from the two points. It passes through the midpoint of the segment connecting the points and is perpendicular to that segment, thus ensuring that any point on this plane maintains equal distances to both endpoints.

Transcript

00:01 So to find the set of points, we're going to rely on a variation of the pythagrum theorem that you guys know of, right? which is a square plus b squared equals z squared.

00:11 The reason why we have three terms on the left hand side is now we're dealing with delta x, delta y, and delta z.

00:18 And that gives us this three -dimensional, this three -dimensional distance, right? and if we want to, you know, find the set of, of points that are equidistant, we need to set this left -hand side equal to both cases, right? so solve it for both cases between this x, y, and z we're looking for, and with a, and then x, y, and z with b.

00:46 And then use both of those cases with this equation and then set those sides equal to one another.

00:52 And then you can just rearrange it and simplify it into a form that you can understand.

00:57 Okay? so let's do that.

00:59 I'm going to keep the formula up there so you can understand where i'm coming from.

01:02 But basically for delta x for a, we're going to do x minus negative 1 because that's the x coordinate.

01:10 So it becomes plus 1 squared plus y minus 5 squared plus z minus 3 squared.

01:20 And that is going to equal the right hand side, which is x minus 6 squared plus 1 .2.

01:30 Y minus two squared plus z plus two squared right because it's minus negative two and then now we can distribute everything out factor everything out so x squared plus two x plus one plus y squared minus 10 y plus 25 plus z squared minus six z um plus nine equal to x squared plus sorry, x squared minus 12x plus 36 plus y squared minus 4 y minus 4 y plus z squared plus 4 z plus 4.

02:23 Okay, it looks right.

02:25 So cross out like terms, which in this case are all of the x terms.

02:30 I mean, all of the squared terms, excuse me.

02:36 And then the next thing we're going to do is move all of the x, y, z terms on one side.

02:45 So if we do that, starting with the x's, we're going to get 14x, right? because the 2 and the negative 12, when we move the negative 12 to the left side.

02:55 And then the next thing is the y...

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