\(
% Arcus cosine.
\def\acos{\cos^{-1}}
% Vector projection.
\def\projection#1#2{{proj_{#1}\left(#2\right)}}
% Vector rejection.
\def\rejection#1#2{{rej_{#1}\left(#2\right)}}
% Norm.
\def\norm#1{{\left\|#1\right\|}}
% Cross product.
\def\cross#1#2{\mathit{cross}\left(#1,#2\right)}
% Dot product.
\def\dot#1#2{{#1 \cdot #2}}
% Magnitude.
\def\mag#1{{\left|#1\right}}
\def\group#1{\left(#1\right)}}
\def\sbgrp#1{\left\{#1\right\}}
\)

If the distance of a point is equal to the radius of the sphere, then we say it is on the sphere.
If the distance of a point is smaller than the radius, then we say the point is inside the sphere.
Otherwise it is outside of the sphere.

If the point is inside or on the sphere, then we say the point and the sphere collide.
Otherwise we say the point and the sphere do not collide.