Given two defined points A and B, what points X exist such that the vectors AX and BX are orthogonal?

Question

For class I have to find: Given two points, say A (1, 1, 1) and B (5, 5 ,5), what is the set of points X (x, y, z) that exist such that the vectors AX and BX are orthogonal?

I know that the points X = A and X = B will work because they will cause the AX and BX vectors to be equal to 0, but I cannot get a definite answer beyond that. I'm currently considering the possibility that the answer is somehow all the points on a circle, but I have no idea how I would go about constructing such a circle.

Any tips that point me in the right direction or show what the scale of my answer (2 points, 3 points, infinite points?) should be would be greatly appreciated.

Final update: Thank you for the concise and useful answers!

I have found that the correct way to answer this question is to take the vectors AX and BX: (1-x, 1-y, 1-z) and (5-x, 5-y, 5-z), find their dot product, and set the resulting equation equal to 0. This gives the equation of a sphere, and all points on the surface of said sphere satisfy the question.

Answer 1

The vector $AX$ is $(x-1,y-1,z-1)$ and the vector $BX$ is $(x-5,y-5,z-5)$. You want them to be orthogonal, so $AX\cdot BX=0$. Compute the dot product and then complete squares.

Answer 2

You should probably look at Thales Theorem.

Answer 3

Certainly there are two more points on the perpendicular bisector of the segment $AB$. (Putting the perpendicular lines in for these two points, you see a square with $AB$ its diagonal.)

Hint: Pick two points on a circle. The central angle subtending the arc bounded by those two points is twice the inscribed angle of the chords from those two points to any third point on the circle. (Thales's theorem is a special case of this.) If the inscribed angle is a right angle, the central angle is ... and the two chords forming that angle together are a ... of the circle.