Plane in two dimention

Introduction

日本語 ver

Today, I prove this theorem.

Plane in two dimention is expressed following.
\[\{x|<x,v> = 0\}\]


however, v is orthogonal vector for plane and not zero vector.

Proof

\[\forall k \in \{x|<x,v> = 0\},\]
k is fulfill this form.
\[<k,v> = 0\]
Now, because k and v in two dimentinal space, each vector express following.
\[k = (k_1,k_2)\]
\[v = (v_1,v_2)\]
Thus, \(<k,v>=k_1v_1 + k_2v_2=0\)

Change this equation.
\[k_2 = -\frac{v_1}{v_2} k_1\]
This equation is plane that slope is \(-\frac{v_1}{v_2}\).
Q.E.D