सिद्ध कीजिए की चार बिंदु ` A,B,C `और D जिनके निर्देशांक क्रमशः `(4,5,1),(0,-1,-1) ,(3,9,4) ` और `(-4,4,4)` है,एक समतलीय है|

Explanation

माना O मूलबिंदु है,तब
` oversetto (OA) =4hati +5hat j+ hat k `
`oversetto (OB) =0hat(i) -hat j- hat k `
` oversetto (OC) =3hati +9hat j +4hatk `
` oversetto (OD) =-4hati +4hatj +4hatk `
अब ` oversetto (AB) =oversetto (OB) -oversetto (OA) =(0hat i -hatj- hat k )- (4hati + 5hat j +hatk ) `
` " "= -4hat i -6hatj - 2hatk =oversetto u ` (माना)
` oversetto (AC) =oversetto (OC) -oversetto (OA) =(3hati +9hat j +4hatk )- (4hati + 5hat j +hatk )`
`" " =-hati+ 4hatj +3hatk =oversetto v ` (माना)
बिंदु A,B,C,D एक समतलीय होंगें यदि सदिश
` oversetto (AD)= oversetto (OD) - oversetto (OA) =(-4hati +4hatj +4hatk )- (4hat i + 5hatj +hat k ) `
` " "=- 8hati -hatj +3hatk = oversetto w` (माना)
बिंदु A,B,C,D एक समतलीय होंगें यदि सदिश `oversetto (AB),oversetto (AC) "तथा" oversetto (AD) "अर्थात" oversetto u,oversetto v` तथा `oversetto w` एक समतलीय होंगें|
अतः ` " "[ oversetto uoversetto voversetto w] =0`
अब `=|{:(,-4,-6,-2),(,-1,4,3),(,-8,-1,3):}|`
` " "=- 4(12+3) +6(-3+24) -2(1+32) `
` " " =-60+126-66=0`
अतः बिंदु `A,B,C,D` एक समतलीय है|