Find an equation of the plane whose points are equidistant from (8,-5,3) and (9,5,9).
The plane would have to pass through the midpoint of the line joining the
two given points and be perpendicular to that line
midpoint = ( (8+9)/2 , (-5+5)/2, (3+9)/2) = (17/2 , 0, 6)
The direction vector, which will become the normal to the plane
= < 9-8, 5+5, 9-3> = <1,10,6>
so the equation of our needed plane is
x + 10y + 6z = c
but (17/2, 0, 6) lies on it, so
17/2 + 0 + 36 = c
17 + 72 = 2c
c = 89/2
x + 10y + 6z = 89/2 is one form of your equation of the plane
another is
2x + 20y + 12z = 89
To find an equation of the plane equidistant from two points, we need to determine the midpoint between the two points and then find the normal vector to the plane. From there, we can write the equation of the plane in vector form.
1. Find the midpoint between the two given points using the midpoint formula:
Midpoint (M) = [(x₁ + x₂) / 2, (y₁ + y₂) / 2, (z₁ + z₂) / 2]
Applying the formula, we get:
M = [ (8 + 9) / 2, (-5 + 5) / 2, (3 + 9) / 2 ]
= [ 17/2, 0, 12/2 ]
= [ 8.5, 0, 6 ]
2. Next, we need to find the normal vector to the plane. The normal vector will be perpendicular to the plane and can be obtained by taking the cross product of the vector between the midpoint and one of the given points.
Let A = (8, -5, 3) be one of the given points.
Let B = (9, 5, 9) be the other given point.
Let N = normal vector to the plane.
N = AB x AM
AB = B - A = (9 - 8, 5 - (-5), 9 - 3) = (1, 10, 6)
AM = M - A = (8.5 - 8, 0 - (-5), 6 - 3) = (0.5, 5, 3)
N = (1, 10, 6) x (0.5, 5, 3)
Taking the cross product, N = (30, -3, -4)
3. Now that we have the midpoint (M) and the normal vector (N), we can write the equation of the plane. The equation of a plane in vector form is:
N · (r - M) = 0
Where N is the normal vector, r is the position vector of an arbitrary point on the plane, and M is the midpoint.
Plugging in the values:
(30, -3, -4) · (r - (8.5, 0, 6)) = 0
Expanding the dot product:
30(r₁ - 8.5) - 3(r₂ - 0) - 4(r₃ - 6) = 0
Simplifying:
30r₁ - 255 - 3r₂ - 4r₃ + 24 = 0
Rearranging terms:
30r₁ - 3r₂ - 4r₃ = 231
Therefore, the equation of the plane is 30x - 3y - 4z = 231.