चूँकि दिय सदिश समतलीय है|
` therefore " " ahati +hatj +hatk =x (hati +bhatj +hatk ) +y( hati +hatj +chatk ) ." " ` जहाँ x ,y अदिश है|
` rArr ahati +hatj +hatk =(x+y) ghati + (bx+ y)hatj + ( x+cy) hatk `
`hati ,hatj ` और ` hatk ` के गुणांकों की तुलना करने पर
` " "x + y =a " "...(1) `
` " " bx + y=1 " "...(2)`
` " " x+ cy =1 " "...(3)`
समी (1 ) से
` " " 1- a =1 -(x+ y) `
समी(2 ) से
` " " bx =1 -y `
`rArr" " b= ( 1-y)/( x) `
` rArr " "1- b =1 -(1-y)/( x )= ( x+ y -1)/( x)`
समी (3 )से
` " "c y =1-x `
`rArr " "c= (1-x)/( y)`
` rArr " "1- c =1 -(1-x)/( y)= (x+ y -1)/( y)`
` therefore " " (1)/( 1-a) +(1)/( 1-b) + (1)/(1-c) =(1)/( 1-( x+y))+( x)/(x+ y -1)+(y)/( x+ y -1)`
` " "= (-1)/( x+y-1)+(x) /( x+y -1)+(y)/(x+ y -1) = ( (x+y-1))/( (x+y-1))=1 `
अतः ` (1)/( 1-a) +(1)/( 1-b) +(1)/(1-c) =1 `
