Let a→=2i^+j^−k^   and  b→=i^+2j^+k^ be two vectors. Consider a vector c→=αa→+βb→,α,β∈ℝ. If the projection of c→   on the vector a→+b→   is 32, , then the minimum value of c→−a→×b→.c→  equals

a¯+b=3i+3j and c=2α+βi+α+2βj+β-αkGiven the projection of c¯ on a¯+b¯ is 32c¯·a¯+b¯a¯+b¯=329α+β=18α+β=2  Considerc¯-a¯×b¯ ·c= c¯2=6α2+6αβ+6β2         since a¯×b¯.c¯=0 as a¯,b¯,c¯ are coplanar =6α2+α2-α+2-α2 =6α2-2α+4 ≤63    since the minimum value of the quadratic expression is 3 =18