Vector A and Vector B intersect, find Vector C that is perpendiculer to Line A and on the same plane as A and B

Question

Just like my drawing, but in 3d Space. I have a point S, a Vector T, and a point O on Vector T.

I want to find a vector A that is perpendicular to Vector T intersect at point O, but at the same time on the same plane as Vector T and Vector OS.

I wonder how would I be able to find it?
Thank you!

Answer 1

First, set your origin at O for simplicity.

Then any vector in the plane defined by OT and OS will be a linear combination of the vectors OT and OS:

$OA = aOT + bOS$

If OA is also perpendicular to OT then

$OA . OT = 0$

$\Rightarrow a(OT.OT) + b(OS.OT) = 0$

$\Rightarrow \frac{b}{a} = - \frac{|OT|^2}{OS.OT}$

So we can have

$OA = (OS.OT)OT - (|OT|^2)OS$

or any scalar multiple of this.