Question
Find an equation of the set of all points equidistant from the points
A(−2, 5, 2) and B(5, 1, −3).
Think of two points in space.
The set of all points equidistant from the points is therefore a plane that separates the two points, right in the middle.
Also the line joining the two points would be perpendicular to the plane.
The mid-point between the two points would evidently lie on the plane.
(1) Find the mid-point between the two given points using the mid-point formula. Let this be p0.
(2) Find the vector AB joining the two given points. Then AB is the normal vector to the plane we're looking for.
Let AB=<a,b,c>
(3) Use the components of the normal vector AB (a,b,c) to form the equation of the required plane:
a(x-x0)+b(y-y0)+c(z-z0)=0
where (x0,y0,z0) is the mid-point P0 between A and B.
If you need more details, please post.
To find the equation of the set of all points equidistant from points A(-2, 5, 2) and B(5, 1, -3), we can follow these steps:
Step 1: Find the midpoint of the line segment AB.
Step 2: Find the direction vector of the line passing through points A and B.
Step 3: Define the equation of the plane that passes through the midpoint and is perpendicular to the line segment AB.
Let's calculate it step-by-step.
Step 1: Finding the midpoint
To find the midpoint M of the line segment AB, use the midpoint formula:
M = ((x1 + x2) / 2, (y1 + y2) / 2, (z1 + z2) / 2)
M = ((-2 + 5) / 2, (5 + 1) / 2, (2 + (-3)) / 2)
M = (3/2, 3/2, -1/2)
So, the midpoint of the line segment AB is M(3/2, 3/2, -1/2).
Step 2: Finding the direction vector
The direction vector of the line passing through points A and B is calculated by subtracting the coordinates of A from B:
v = B - A
v = (5, 1, -3) - (-2, 5, 2)
v = (5 + 2, 1 - 5, -3 - 2)
v = (7, -4, -5)
So, the direction vector of the line segment AB is v(7, -4, -5).
Step 3: Defining the equation of the plane
To find the equation of the plane passing through the midpoint M, perpendicular to the direction vector v, we can use the following equation:
Ax + By + Cz = D
Substituting the values of M(3/2, 3/2, -1/2) and v(7, -4, -5) into the equation, we get:
7(x - 3/2) - 4(y - 3/2) - 5(z + 1/2) = 0
Multiplying out terms and simplifying, we have:
7x - 21/2 - 4y + 6 + 5z + 5/2 = 0
7x - 4y + 5z = 5/2 + 21/2 - 6
7x - 4y + 5z = 10
Therefore, the equation of the set of all points equidistant from points A(-2, 5, 2) and B(5, 1, -3) is:
7x - 4y + 5z = 10
To find an equation of the set of all points equidistant from the points A(-2, 5, 2) and B(5, 1, -3), we can use the concept of the midpoint and distance formula.
Step 1: Find the midpoint of the line segment AB.
To find the midpoint of the line segment AB, we use the formula:
Midpoint = ((x1 + x2) / 2, (y1 + y2) / 2, (z1 + z2) / 2)
Let's substitute the coordinates of A and B into the formula:
Midpoint = ((-2 + 5) / 2, (5 + 1) / 2, (2 + (-3)) / 2)
= (3/2, 6/2, -1/2)
= (3/2, 3, -1/2)
= (1.5, 3, -0.5)
So the midpoint of the line segment AB is (1.5, 3, -0.5).
Step 2: Find the distance from the midpoint to either of the points A or B.
We can use the distance formula to find the distance from the midpoint to either of the points A or B. The distance formula is given by:
Distance = sqrt((x2 - x1)^2 + (y2 - y1)^2 + (z2 - z1)^2)
Let's calculate the distance from the midpoint to point A:
Distance = sqrt((1.5 - (-2))^2 + (3 - 5)^2 + (-0.5 - 2)^2)
= sqrt((3.5)^2 + (-2)^2 + (-2.5)^2)
= sqrt(12.25 + 4 + 6.25)
= sqrt(22.5)
= 4.743
The distance from the midpoint to point A is approximately 4.743.
Step 3: Write the equation of the set of all points equidistant from A and B.
Since the set of all points equidistant from A and B is a sphere, we can write the equation in the form:
(x - h)^2 + (y - k)^2 + (z - l)^2 = r^2
where (h, k, l) represents the coordinates of the midpoint and r is the radius (distance from the midpoint to either A or B).
In this case, the coordinates of the midpoint are (1.5, 3, -0.5), and the radius is 4.743. Therefore, the equation of the set of all points equidistant from A and B is:
(x - 1.5)^2 + (y - 3)^2 + (z + 0.5)^2 = 4.743^2
or
(x - 1.5)^2 + (y - 3)^2 + (z + 0.5)^2 = 22.5
This equation represents the set of all points that are equidistant from A(-2, 5, 2) and B(5, 1, -3).