Let a , b and c be three vectors having magnitude 1,1 and 2 respectively. If a ( a c )+ b = 0 , then the acute angle between a and c is

Step-by-step solution

Given, | a |=1,| b |=1 and | c |=2 aligned & Also, a ( a c )+ b = 0 & ( a c ) a -( a a ) c + b = c & ( a c ) a - c + b = 0 [ a a =| a |^2=1 ] & ( a c ) a - c =- b & |( a c ) a - c |=|- b | & |( a c )^ a - c |^2=| b |^2 & |( a c ) a |^2+| c |^2-2 ( a c ) a c =| b |^2 & ( a c )^2| a |^2+| c |^2-2( a c )( a c )=| b |^2 & ( a c )^2 | a |^2-2 +| c |^2=| b |^2 aligned aligned & -( a c )^2+4=1 [ | ~b |^2=1,| c |^2=4 ] & ( a c )^2=3 & a c = 3 & | a || c | = 3 aligned where is an acute angle between a and c = 3 2 = 6