Find an equation of the set of all points equidistant from the points
A(−2, 6, 3) and B(5, 3, −3)
To find an equation of the set of all points equidistant from two given points A(-2, 6, 3) and B(5, 3, -3), you can follow these steps:
1. Start by finding the midpoint M of points A and B. The midpoint formula is:
M = ((x₁ + x₂)/2, (y₁ + y₂)/2, (z₁ + z₂)/2)
Plugging in the coordinates of A and B, we get:
M = ((-2 + 5)/2, (6 + 3)/2, (3 + -3)/2)
= (3/2, 9/2, 0)
2. Next, find the distance between points A and M using the distance formula. The distance formula is:
d = √((x₂ - x₁)² + (y₂ - y₁)² + (z₂ - z₁)²)
Plugging in the coordinates of A and M, we get:
d = √((3/2 - (-2))² + (9/2 - 6)² + (0 - 3)²)
= √(49/4 + 1/4 + 9)
= √(49 + 1 + 36)
= √(86)
3. Now, we can write the equation of the set of points equidistant from A and B. Since these points are equidistant, the distance between any point on the set and A should be equal to the distance between that point and B. Let's call the unknown point (x, y, z). The equation can be written as:
√((x - (-2))² + (y - 6)² + (z - 3)²) = √(86)
Simplifying the equation further, we get:
(x + 2)² + (y - 6)² + (z - 3)² = 86
Therefore, the equation of the set of all points equidistant from A(-2, 6, 3) and B(5, 3, -3) is (x + 2)² + (y - 6)² + (z - 3)² = 86.
you know it will be a plane normal to the vector <7,-3,-6> passing through (3/2,9/2,0).
I'm sure you've covered that.