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
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
Hi Aileys.
It's not a coincidence. In vector, $\displaystyle \left(\begin{array}{cc}x\\y\\z\end{array}\right)$ usually denoted by $\displaystyle \vec r$
So, $\displaystyle ax+by+cz=d$ can be written as :
$\displaystyle \left(\begin{array}{cc}x\\y\\z\end{array}\right) \cdot \left(\begin{array}{cc}a\\b\\c\end{array}\right) = d$
$\displaystyle \vec r \cdot \left(\begin{array}{cc}a\\b\\c\end{array}\right) = d$
That's a general equation of plane where $\displaystyle \left(\begin{array}{cc}a\\b\\c\end{array}\right)$ is the normal vector