# Math Help - Equation of plane & normal vector

1. ## Equation of plane & normal vector

OK so if the equation of a plane is ax + by + cz = d you can just read off that the normal vector is (a,b,c)... but why is this? Is there a geometric kind of argument for why it's like that, or is it just a handy co-incidence?

Thanks

2. Hi Aileys.

It's not a coincidence. In vector, $\left(\begin{array}{cc}x\\y\\z\end{array}\right)$ usually denoted by $\vec r$

So, $ax+by+cz=d$ can be written as :

$\left(\begin{array}{cc}x\\y\\z\end{array}\right) \cdot \left(\begin{array}{cc}a\\b\\c\end{array}\right) = d$

$\vec r \cdot \left(\begin{array}{cc}a\\b\\c\end{array}\right) = d$

That's a general equation of plane where $\left(\begin{array}{cc}a\\b\\c\end{array}\right)$ is the normal vector