Let ( a =2 i - j +3 k , ~b =3 i -5 j + k ) and ( c ) be a vector such that ( a c = c b ) and (( a + c ) ( b + c )=168 ). Then the maximum value of (| c |^2 ) is :

Step-by-step solution

aligned & a =2 i - j +3 k & ~b =3 i -5 j +3 k & a c = c b & a c + b c =0 & ( a + b ) c =0 & c = ( a + b ) & c = (5 i -6 j +4 k ) . .(1) & | c |^2= ^2(25+36+16) & | c |^2=77 ^2 & ( a + c ) ( b + c )=168 & a ~b + a c + c ~b +| c |^2=168 aligned 14+ c ( a + b )+77 ^2=168 using equation (1) aligned & |5 i -6 j +4 k |^2+77 ^2=154 & 77 +77 ^2-154=0 & ^2+ -2=0 & =-2,1 aligned Maximum value of | c |^2 occurs when =-2 aligned & | c |^2=77 ^2 & =77 4 & =308 aligned