Find the vector equation of the plane passing through the point `A(1, 0, 1), B(1,-1,1) and C (4,-3,2)`

Explanation

Let `bara, barb, barc` be the position vectors of points A, B, C respectively.
`:. " " bara=hati+hatk, barb=hati-hatj+hatk, barc`
` = 4 hati-3hatj+2hatk.`
` :. bar(AB) = barb - bara = (hati-hatj+hatk) - (hati+hatk)`
` = - hatj`
` bar(AC) = barc-bara = (4hati-3hatj+2hatk) - (hati+hatk)`
` = 3 hati-3hatj+hatk`
` :. bar(AB) xx bar(AC) = |(hati,hatj,hatk),(0,-1,-0),(3,-3,1)|`
`= hati (-1+0) - hatj(0) + hatk(0+3)`
` = - hati + 3 hat k.`
Therefore , vector equation of the required plane is , `barr*(bar(AB) xx bar(AC)) = bara(bar(AB) xx bar(AC))`
` rArr barr * (-hati+3 hatk)=(hati+hatk) *(-hati+ 3 hatk)`
`rArr barr * (-hati+3 hatk) = 1 (-1) + 1 (3)`
` rArr barr*(-hati+3hatk) = - 1 + 3 =2`
` rArr barr * (-hati+ 3 hatk) = 2`
This is the required vector equation of the plane.