Answer to Question #163677 in Linear Algebra for Nikhil Rawat

Question #163677

Find the vector equation of the plane determined by the points (1,-2,1),(1,0,1) and (1,-1,1). Also check whether (1/2,1/2,1/2) lies on it.


1
Expert's answer
2021-02-16T15:46:27-0500

Solution.

If the coordinates of the three points A(x1;y1;z1),A(x_1;y_1;z_1), B(x2;y2;z2)B(x_2;y_2;z_2) and C(x3;y3;z3)C(x_3;y_3;z_3) , which lie on the plane, the plane equation can be found by the following formula


xx1yy1zz1x2x1y2y1z2z1x3x1y3y1z3z1=0.\begin{vmatrix} x-x_1 & y-y_1 & z-z_1 \\ x_2-x_1 & y_2-y_1 & z_2-z_1 \\ x_3-x_1 & y_3-y_1 & z_3-z_1 \\ \end{vmatrix}=0.

We will get


x1y+2z1020010=0,\begin{vmatrix} x-1 & y+2 & z-1 \\ 0 & 2 & 0 \\ 0 & 1 & 0 \\ \end{vmatrix}=0,

from here


(x1)0+(y+2)0+(z1)0=0,or0x+0y+0z+0=0.(x-1)•0+(y+2)•0+(z-1)•0=0, \newline\text{or}\newline 0\cdot x+0\cdot y+0\cdot z+0=0.

n=(000)\vec{n}=\begin{pmatrix} 0 \\ 0 \\ 0 \end{pmatrix} - normal vector.

The vector equation of the plane:


<n,r>=a,<\vec{n},\vec{r}>=a,

where n={A;B;C}={0;0;0},\vec{n}=\{A;B;C\}=\{0;0;0\}, r={x,y,z},\vec{r}=\{x,y,z\}, and a=D=0.a=-D=0.

So,


(000)×(xyz)=0\begin{pmatrix} 0 & 0 & 0 \end{pmatrix} ×\begin{pmatrix} x\\ y\\ z \end{pmatrix}=0

is the vector equation of the plane. It is not defined. The point (1/2;1/2;1/2) lies on it.


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