Answer to Question #195524 in Linear Algebra for Nikhil

Question #195524

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
2021-05-24T02:01:02-0400

x(1)y(2)z11(1)0(2)111(1)1(2)11=0\begin{vmatrix} x-(-1) & y-(-2) & z-1 \\ 1-(-1) & 0-(-2) & 1-1 \\ 1-(-1) & -1-(-2) & 1-1 \end{vmatrix}=0


x+1y+2z1220210=0\begin{vmatrix} x+1 & y+2 & z-1\\ 2 & 2 & 0\\ 2 & 1 & 0 \end{vmatrix}=0


(z1)[24]=0(z-1)[2-4]=0

z=1z=1


r.(0i+0j+1k)=1\vec r.(0\vec i+0\vec j+1\vec k)=1


Put the point (x = 1/2. y = 1/2, z = 1/2) on the equation of plane i.e

0x+0y+z=10x+0y+z=1

12=1\dfrac{1}{2}\cancel=1

Therefore, the point (1/2,1/2,1/2) does not lie on the plane


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