Question

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

Answer 1

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

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

$(z-1)[2-4]=0$

$z=1$

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

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

$0x+0y+z=1$

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

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