If a , b , c are three vectors with magnitudes 3 , 1,2 respectively, such that a ( a c )+3 b = 0 , if is the angle between a and c , then ^2 is

Step-by-step solution

aligned & Given | a |= 3 ,| b |=1,| c |=2 & a ( a c )+3 b = 0 & ( a c ) a -( a a ) c +3 b = 0 & ( a c ) a -3 c =-3 b [ a a =| a |^2=3 ] & |( a c ) a -3 c |=|-3 b | & |( a c )⁻ a -3 c |^2=|-3 b |^2 & |( a c )⁻ |^2+9| c |^2-6 ( a c )( a c ) =9| b |^2 & ( a c )^2| a |^2+9| c |^2-6( a c )( a c )=9| b |^2 & ( a c )^2 (| a |^2-6 )+9| c |^2=9| b |^2 & ( a c )^2(3-6)+9(2)^2=9(1)^2 aligned aligned & -3( a c )^2=-27 & ( a c )^2=9 & a c = 3 & | a || c | = 3 & ( 3 )(2) = 3 & = 3 2 & = 2 3 & ^2 = 4 3 aligned