If , `vecx, vecy` are two non-zero and non-collinear vectors satisfying
`[(a-2)alpha^(2)+ (b-3)alpha+c] vecx + [(a-2)beta+c]vecy + [(a-2)gamma^(2)+(b-3)gamma+c] (vecx xx vecy) = 0, " where " alpha, beta, gamma` are three distinct distinct real numbers, then find the value of `(a^(2) + b^(2) + c^(2) - 4)`

Answer

9

Explanation

Since `vecx and vecy` are non- collinear vectors, therefore, `vecx, vecy and vec xx vecy` are non- coplanar vectors.
`[(veca -2)alpha^(2)+(b+3)alpha+c]+[(a-2)beta^(2)+ (b-3)betabeta+c] vecy +[(a -2)gamma^(2)+(b-3)gamma+c] (vecx xx vecy)=0`
Cofficient of each vector `vecx, vecy and vecx xx vecy` is zero.
`(a-2)alpha^(2)+(b-3)alpha+c=0`
`(a-2)beta^(2)+(b-3)beta+c=0`
` (a-2)gamma^(2)+(b-3)beta+c=0`
The above three equations will satisfiy if the coefficients of `alpha,beta and gamma` are zero because `alpha, beta and gamma` are three distinct real numbers
`a-2=0 or a =2`
`b-3=0 or b=3 and c=0`
`a^(2)+b^(2) +c^(2) = 2^(2)+3^(2)+0^(2)=4+9=13`