Finding the missing vertex $(x,y,z)$ of a rectangle whose other vertices are defined.

Question

How do I find the missing fourth vertex $D$ of a rectangle, which has three vertices defined?

The Equation of the plane being $ax+by+cz+d=0$

Where,

$a = (By-Ay)(Cz-Az)-(Cy-Ay)(Bz-Az)$

$b = (Bz-Az)(Cx-Ax)-(Cz-Az)(Bx-Ax)$

$c = (Bx-Ax)(Cy-Ay)-(Cx-Ax)(By-Ay)$

$d = -(aAx+bAy+cAz)$

Answer 1

I found the following picture that would help clarify my question. To find W, add T and V and Subtract U.

Equations

1) Main Equation

$w = t + v - u$

2) 3D Point Equations

$(x,y,z)\mathbf{_{w}} = (x,y,z)\mathbf{_{t}} + (x,y,z)\mathbf{_{v}} - (x,y,z)\mathbf{_{u} }$

3) Individual 3D Coordinate Equations

$x\mathbf{_{w}} = x\mathbf{_{t}} + x\mathbf{_{v}} - x\mathbf{_{u} }$

$y\mathbf{_{w}} = y\mathbf{_{t}} + y\mathbf{_{v}} - y\mathbf{_{u} }$

$z\mathbf{_{w}} = z\mathbf{_{t}} + z\mathbf{_{v}} - z\mathbf{_{u} }$

Finding Missing 4th Vertex of a Parallelogram

Image Courtesy of: http://www.goldensoftware.com/forum/viewtopic.php?f=1&t=8174

Extending this to a 3D would mean just performing the same operation on the Z-Axis.