Math Help - 3D vectors

1. 3D vectors

Given points A and B in 3-dimensional space, describe the solutions to
−→ −→
P A • P B = 0.

2. Re: 3D vectors

Hello, romean2015!

$\text{Given points }A\text{ and }B\text{ in 3-space, describe the solutions to: }\overrightarrow{PA}\cdot\overrightarrow{PB}=0$

Consider a sphere with $AB$ as its diameter.
Consider any point $P$ on this sphere.
Draw chords $PA$ and $PB.$
Code:
              * * *
*           *  P
*               o
*                 *

*                   *
A o - - - - * - - - - o B
*                   *

*                 *
*               *
*           *
* * *
$\text{Since }\angle APB = 90^o,\,\text{ then: }\,\overrightarrow{PA}\cdot\overrightarrow{PB} \,=\,0$

$\text{Therefore, the locus of point }P\text{ is the sphere with diameter }AB.$