Vectors in three-dimensional space: locus of a variable point The fixed point B has position vector $b$ relative to a fixed point $O$. A variable point $M$ has position vector $m$ relative to $O$. Find the locus of M if m $\cdot$ (m – b)=0.
I am told the answer is derived as such:

Can somebody explain the solution to me? I've tried but can't seem to understand...thanks!

If $\vec{OM}=(x,y,z)$ and $\vec{OB}=(a,b,c)$ then
$$\vec{OM}.\vec{BM}=0\implies$$
$$x(x-a)+y(y-b)+z(z-c)=0 \implies$$
$$(x-\frac{a}{2})^2+(y-\frac{b}{2})^2+(z-\frac{c}{2})^2=\frac{a^2+b^2+c^2}{4}$$
the point M is in the sphere of center $(\frac{a}{2},\frac{b}{2},\frac{c}{2})$ and radius
$R=\frac{\sqrt{a^2+b^2+c^2}}{2}$.
