How to simplify this dot expression.

Question

I feel I have a fair understanding of the vector equation of a plane, however I am stumbling on manipulating the following equation to t: Equation Post I need assistance in understanding how the author simplified the plane equation to the solution t, specifically how to move the dot operation from one side to the other or how to factor t out to the other side. Thank you.

$Ray P = Po + tv$

$Plane Eq= P\cdot N + d = 0$

Substituting Plane Eq with Ray P:

$$PlaneEq= (Po + tv) N + d = 0$$ Here is where I lose the author: $$t = -(Po\cdot N+d)/V \cdot N$$ How do I go from PlaneEq to t, step by step? In particular how do I move the dot expression within the equation??

Answer 1

$$(P_0 +tv)\cdot N +d=0$$ $$P_0\cdot N +tv\cdot N +d=0$$ $$tv\cdot N=-P_0\cdot N- d$$ $$t(v\cdot N)=-(P_0\cdot N+d)$$ $$t=-(P_0\cdot N+d)/(v\cdot N)$$

Note $(tv)\cdot N$ is the same as $t(v\cdot N)$.

Answer 2

I think if we take it step by step the problem won't be so difficult.

$$(P_0 +tv) \cdot N +d = 0 \implies P_0 \cdot N +t v \cdot N + d = 0$$ $$\implies t (v \cdot N) =-P_0 \cdot N- d \implies t = -(P_0 \cdot N + d)/ (v \cdot N)$$

Hope that helps.