How to find set of points to satisfy a vector paralel to a known vector?

Question

I am given two points: $C(4,-2,3)$ and $U(1,0,-2)$.

How do I find the set of points $P(x,y,z)$, knowing that $\vec{UC} \cdot \vec UP=0$?

I know that $\vec{UC}$ and $\vec{UP}$ should be parallel, but how do I develop this idea?

Answer 1

You said " I know $\vec{UC}$ and $\vec{UP}$ should be parallel ". The fact is they need not be. What made you think so? back to your question, @Don Thousand already addressed it. I just brush it up a bit. Let $P = (x,y,z) \implies \vec{UP} = <x-1,y,z+2>, \vec{UC} = <3,-2,5>$. Use the definition of dot product for $2$ perpendicular vectors, namely: if $ \vec{X} = <x_1,y_1,z_1>, \vec{Y} = <x_2,y_2,z_2>$ then $\vec{X}\cdot \vec{Y} = 0 \iff x_1x_2+y_1y_2+z_1z_2 = 0$. Can you take it from here...?