The 3-D Coordinate System — Question 6

Question 6

Describe geometrically the set of all points $P=(x,y,z)$ equidistant from $A=(2,-1,3)$ and $B=(-4,5,1)$. Derive a Cartesian equation for the set and show that the midpoint of $AB$ belongs to it.

Question 6 – Solution

Strategy Equate squared distances. Quadratic terms cancel, revealing the perpendicular-bisector surface.

Derivation Set $$(x-2)^2+(y+1)^2+(z-3)^2=(x+4)^2+(y-5)^2+(z-1)^2.$$ After expanding and simplifying, $\boxed{3x-3y+z+7=0}$.

Interpretation This is the plane perpendicular to $\overrightarrow{AB}=\langle-6,6,-2\rangle$ through the midpoint $M=(-1,2,2)$. Indeed, $3(-1)-3(2)+2+7=0$.

Verification Interchanging $A$ and $B$ does not change the locus, and every point in the plane has equal squared distance to the endpoints.