Chapter 11, Problem 65RE

To determine

The general form of the equation of the plane with the given characteristics.

Answer to Problem 65RE

The general equation of the plane is z=2.

Explanation of Solution

Given:

Passes through (5,3,2) and is parallel to the x-y plane.

Assuming the points $\left(x_1,y_1,z_1\right),\left(x_2,y_2,z_2\right),\left(x_3,y_3,z_3\right)$ then the equation of the plane passing from these lines are given as:

  $a\left(x-x_1\right)+b\left(y-y_1\right)+c\left(z-z_1\right)=0$

Considering the point (1,2,3) and the xy-plane:

For xy plane z=0.

Hence, the vector parallel to the plane is $\langle 0,0,1\rangle$ .

Therefore,

  $\langle 0,0,1\rangle =\langle a,b,c\rangle$

The equation of the plane is given as:

  $\begin{array}{c}a\left(x-x_1\right)+b\left(y-y_1\right)+c\left(z-z_1\right)=0\\ 0\left(x-5\right)+0\left(y-3\right)+1\left(z-2\right)=0\\ z-2=0\\ z=2\end{array}$

Therefore, the general equation of the plane is z=2.