Q 69 - Calculus

Question

For constants a, b, and c, a function of two variables of the form $f(x, y)=ax+by+c$ is called a linear function of two variables. Show that the graph of the linear function $f(x, y)=ax+by+c$ is a plane with normal vector (a, b, −1) containing the point (0, 0,c). 

Step-by-Step Solution

Answer

We proved the f is a plane with normal vector (a, b, -1) containing the point (0, 0 ,c)

Step 1: Given information

We are given a function of two variables 

$f(x,y)=ax+by+c$

Step 2: Explanation

We have to show that f is a plane with a given normal vector and a coordinates

Let the normal vector be $(a,b,-1)$

Let $r=(x,y,0)$

And $r_0=(0,0,c)$

We know that

$n·(r-r_0)$ is the equation of the plane

Substituting the values we get,

$\begin{array}{rcl}(a,b,-1)·\left(\right(x,y,0)-(0,0,c\left)\right)& =& (a,b,-1)(x,y,-c)\\ & =& ax+by+c\\ & =& f(x,y)\end{array}$

Hence f the function of two variables is a plane with a normal vector (a, b,-1) containing the point (0, 0 ,c)