Find a vector equation for the line through the point $(a,b,c)$ which is normal to the plane $\cdots$

Question

Find a vector equation for the line through the point $(a,b,c)$ which is normal to the plane through $P(x_1,y_1,z_1), Q(x_2,y_2,z_2) $and $R(x_3,y_3,z_3)$

What would the process be to answer a question like this?

Answer 1

First of all you need the direction of the line. Since it is perpendicular to the plane it is perpendicular to vector $\vec{PQ}$ and $\vec{PR}.$ Thus, the vector $$\vec{PQ}\times\vec{PR}$$ gives you the direction of the line. Since the line contains the point $(a,b,c)$ you have the equation

$$\frac{x-a}{u}=\frac{y-b}{v}=\frac{z-c}{w},$$ where $(u,v,w)=\vec{PQ}\times\vec{PR}.$ Or, since you ask for the vector equation,

$$(x,y,z)=(a,b,c)+\lambda(u,v,w).$$