Let's say I have a plane given by the equation 2x - y + z = 0
What does the vector <2,-1,1> represent with regards to this plane? Is it the slope, the gradient? Is it perpendicular to the surface or does it go along the surface?
What is a normal vector? Is the normal vector <2,-1,1> or is it something else entirely? Is the normal vector the gradient? I am confused about what these terms mean geometrically.
Thank you!
On
It is the vector normal to the plane. It means that it is orthogonal to any vector on this plane. For a easier approach, any line on the plane is at right angle with the normal of the plane if they cut each other. Here, the gradient is the normal to the plane. However, the gradient is the vector that points to the direct that varies the most for a multivariable function (general). A plane $ax+by+cz=0$ has a gradient that is $(a,b,c)$ which is, luckily always the gradient, but it's just a "coincidence".
Recall that by dot product
$$(a,b,c)\cdot(x,y,z)=ax+by+cz=0$$
then the vector $n=(a,b,c)$ is orthogonal to any point $OP=(x,y,z)$ on the plane.
Note also that more in general, since $ax+by+cz=d$ is obtained as a simple translation of $ax+by+cz=0$ the vector $n$ is also orthogonal to $ax+by+cz=d$.