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

00:02 Let's find the equation that describes all points that are equidistant between points a, negative 1, 5, 3, and b 6, 2, negative 2.

00:15 What's that going to look like? well, if we're in two dimensions, and i have two points, we'll just call this one a and b, the points that are the same distance between a and b fall along this line, which is the perpendicular bisector of the point.

00:39 The line segment from a to b.

00:45 And all of the points on this line are the same distance from a as they are to b.

01:02 In three dimensions, we can imagine viewing this line from many perspectives.

01:09 Imagine holding down points a and b and spinning it around coming in and out of the writing surface along that axis.

01:17 That's the line segment between them.

01:20 That's going to give you a whole.

01:22 Whole plane of lines that are equidistant between a and b.

01:30 So in three dimensions, we are looking at a plane that all of these lines fall in such that any point, let's call it p, on that plane.

01:52 It's going to be the same distance away from a as it is from b.

01:58 Ap equals bp.

02:03 And that's going to be the basis for coming up with our equation.

02:07 We're going to assign p, coordinates x, y, and z because it can be any point on this plane.

02:17 So whatever point we're referring to is going to have coordinates x, y, and z.

02:21 We're going to find the distance from a to p using the distance formula.

02:27 We're going to find the distance from b to p and set them equal to each other.

02:32 And that's going to give us the equation that describes all of these points that are the same distance from a as they are to b.

02:43 So let's apply the distance formula using x, y, and z for point p, and the coordinates negative 153 for point a, and 6 to negative 2 for point b.

02:55 So the distance from a to p is the square root of subtracting x coordinates, x minus negative 1 squared plus y minus 5 squared plus z minus 3 squared the distance from b to p subtracting x coordinates x minus 6 square plus y minus 2 square plus z minus negative 2 squared i'm going to observe that if ap is equal to bp, then they're squares.

03:56 Must be equal to each other.

03:59 And i want to do that to kind of simplify the algebra that i'm about to do.

04:03 That'll get rid of my square roots.

04:05 If i look at the squares of these distances and set them equal to each other.

04:12 We're not going to have to worry about any extraneous roots being introduced because the radicans inside here can never be negative.

04:20 So we're not and have the possibility that one was negative and one was positive, adding new solutions...

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