Question

Why are the coefficients from the plane ax+by+cz=d used to for the normal vector ?

Answer 1

I am seeking an explanation for the method of finding the normal vector of plane.

Answer 2

A normal vector is any vector that starts at some point on the plane to a point perpendicular to the plane.

Answer 3

right

Answer 4

I know this but I am wondering how this method works. How can you just pull the coefficients off and call it the normal vector? What is the logic that justifies this...?

Answer 5

Those coefficients form a vector that spans R.

Answer 6

So for any values of x y and z you will get a vectors that are oriented in the same direction

Answer 7

The coefficients form the vector that defines the plane so they aren't really pulled out to define a vector, they form a vector in that plane.

Answer 8

Pondering..

Answer 9

The vector formed by those coefficients is only normal if its dot product with the vector PQ (P in plane Q out of plane) is zero

Answer 10

It may be best to derive the formula for you.

In order to define a plane you need two points in the plane, and a vector perpendicular to it.
Let us pick a point and call it Po=(xo,yo,zo). Call ro=<xo,yo,zo> be the vector that points from the origin to that point, and call the normal vector n.
Let any given point in the plane be represented by P=(x,y,z),amd the vector that points to it be r=<x,y,z>

That means the vector r-ro is in the plane.
If the vector r-ro is in the plane, then the it is perpendicular to the normal vector, which means that their dot product is zero.\[n\cdot(P-P_o)=0\]

Answer 11

sorry, formula should be\[n\cdot(r-r_o)=0\]\[n\cdot r=n\cdot r_o\]let n=<a,b,c>
we then have\[<a,b,c>\cdot<x,y,z>=<a,b,c>\cdot<x_o,y_o,z_o>\]\[ax+by+cz=ax_o+by_o+cz_o\]because the quantity on the right is a constant we can call it d, and we get\[ax+by+cz=d\]now that you know how we derived the formula, it should make more sense that we can run it backwards and say that the normal vector is just composed of the coefficients on the variables.