Implicitly Defined Surface (Part 2)

Geometry Level 3

In a 3D coordinate system, vector $\vec{P}$ is a reference vector. Consider a surface defined by the set of points $\vec{Q}$ such that:

$\large{\vec{Q} \cdot (\vec{P} - \vec{Q}) = -|\vec{Q}}|^2$

What type of surface is this?

Note: The symbol " $\large{\cdot}$ " denotes the dot product operation. $|\vec{Q}|$ denotes the absolute value of $\vec{Q}$ .

Ellipsoid Finite Sphere Plane Paraboloid

1 solution

Michael Mendrin
Feb 15, 2017

Let $P=(a,b)$ and $Q=(x,y)$ , where $x,y$ varies. Then

$Q \cdot (P-Q)=Q \cdot P-Q \cdot Q=-|Q|^2=-Q \cdot Q$

so that

$Q \cdot P=0$

which defines a plane perpendicular to $P$ .