Let a =2 i + j + k , b =- i + k , c = j - k , where and are integers and =-6 . Let the values of the ordered pair ( , ) , for which the area of the parallelogram of diagonals a + b and b + c is 21 2 , be ( ₁, ₁ ) and ( ₂, ₂ ) . Then ₁^2+ ₁^2- ₂ ₂ is equal to

Step-by-step solution

aligned & Area of parallelogram .= 1 2 , d ₁ d ₂ & ~A = 1 2 |( a + b ) ( b + c )|= 21 2 & so, a + b = i + j +2 k & b + c =- i + j & ( a + b ) ( b + c )= | array ccc i & j & k 1 & & 2 -1 & & 0 array | aligned aligned & = i (-2 )- j (2)+ k ( + ) & |( a + b ) ( b + c )|= 4 ^2+4+( + )^2 = 21 & 4 ^2+4+ ^2+ ^2+2 =21 & ^2+5 ^2-12=17 & ^2+5 ^2=29 aligned and =-6 and given _ i are integers so, =-3, =2 or aligned & =3, =-2 & ( ₁, ₁ )=(-3,2) & ( ₂, ₂ )=(3,-2) & ₁^2+ ₁^2- ₂ ₂=9+4+6=19 aligned