The set of all points equidistant from (0, 1, 5) and 5, -1, 3) is a plane. Find the equation.
Let P = (0, 1, 5) and Q = (5, -1, 3) be the given points. The set of all points equidistant from P and Q lies on the perpendicular bisector of the line segment PQ.
To find the equation of the perpendicular bisector, we need to find its midpoint and its direction.
The midpoint M of PQ is given by the average of the coordinates of P and Q:
M = ((0+5)/2, (1+(-1))/2, (5+3)/2) = (2.5, 0, 4)
Now, let's find the direction vector of PQ. The direction vector is obtained by subtracting the coordinates of P from Q:
d = Q - P = (5, -1, 3) - (0, 1, 5) = (5, -2, -2)
To find the direction vector of the perpendicular bisector, we take the perpendicular vector to d. We can do this by swapping the x and y coordinates and changing the sign of one of them.
So, the direction vector of the perpendicular bisector is (-2, 5, -2).
Finally, using the coordinates of the midpoint M and the direction vector (-2, 5, -2), we can write the equation of the plane as:
-2(x - 2.5) + 5(y - 0) - 2(z - 4) = 0
which simplifies to:
-2x + 5y - 2z + 3 = 0
Therefore, the equation of the plane containing all the points equidistant from P and Q is -2x + 5y - 2z + 3 = 0.
To find the equation of the plane that contains all the points equidistant from (0, 1, 5) and (5, -1, 3), we can use the concept of the midpoint between two points.
Let's denote the midpoint between (0, 1, 5) and (5, -1, 3) as M. The midpoint is found by averaging the coordinates of the two points.
Midpoint M = [(0+5)/2, (1-1)/2, (5+3)/2]
= [2.5, 0, 4]
Now, let's find the vector from the midpoint M to one of the given points, for example, to (0, 1, 5).
Vector V = (0, 1, 5) - M
= (0 - 2.5, 1 - 0, 5 - 4)
= (-2.5, 1, 1)
The equation of a plane can be expressed as Ax + By + Cz + D = 0, where A, B, C, and D are constants.
Now, we can use the vector V to determine the coefficients A, B, and C of the equation of the plane.
A = -2.5
B = 1
C = 1
To determine the constant D, we can substitute the coordinates (2.5, 0, 4) (the midpoint of the two points) into the equation Ax + By + Cz + D = 0.
-2.5(2.5) + 1(0) + 1(4) + D = 0
-6.25 + 4 + D = 0
D = 2.25
Thus, the equation of the plane that contains all the points equidistant from (0, 1, 5) and (5, -1, 3) is:
-2.5x + y + z + 2.25 = 0.