Write an equation for the plane that contains the point A = (4,5,-3) and that is perpendicular to the line through B = (5,-2,-2) and C = (7,1,4)
one way to write the equation of a plane is:
$\displaystyle ax + by + cz + d = 0$
here, $\displaystyle \bold{n} = \left< a,b,c \right>$ is the normal vector to the plane. here, your normal vector is the vector $\displaystyle \overrightarrow {BC}$, do you see why?
we begin with $\displaystyle a(x - x_0) + b(y - y_0) + c(z - z_0) = 0$
where $\displaystyle (x_0,y_0,z_0)$ is a point the plane passes through. then we expand and simplify