Meeting Point of a Line and a Plane

Theorem 6.23. The point where the line $\vec r=\vec a+t\vec b$ meets the plane $\vec r\cdot\vec n=p$ (assuming the line is not parallel to the plane, i.e. $\vec b\cdot\vec n\ne0$) has position vector

Proof. Let $\vec u$ be the position vector of the meeting point; it satisfies both the line's equation, $\vec u=\vec a+t_1\vec b$ for some parameter $t_1$, AND the plane's equation, $\vec u\cdot\vec n=p$. Substituting the first into the second: $(\vec a+t_1\vec b)\cdot\vec n=p\ \Rightarrow\ \vec a\cdot\vec n+t_1(\vec b\cdot\vec n)=p\ \Rightarrow\ t_1=\dfrac{p-\vec a\cdot\vec n}{\vec b\cdot\vec n}$ (valid precisely because $\vec b\cdot\vec n\ne0$). Substituting this $t_1$ back into $\vec u=\vec a+t_1\vec b$ gives the boxed formula. …