Let a , b , c be three vectors having magnitudes 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 ( a c ) + b = 0 Using the vector triple product expansion: ( a c ) a - ( a a ) c + b = 0 Since | a | = 1 , a a = 1 . ( a c ) a - c + b = 0 b = c - ( a c ) a Squaring both sides: | b |^2 = | c |^2 + ( a c )^2 | a |^2 - 2( a c )( a c ) | b |^2 = | c |^2 - ( a c )^2 Substituting the given magnitudes | a | = 1 , | b | = 1 , and | c | = 2 : 1^2 = 2^2 - (| a || c | )^2 1 = 4 - (1 2 )^2 1 = 4 - 4 ^2 4 ^2 = 3 ^2 = 3 4 = 3 2 For the acute angle, = 3 2 = 6 Answer: 6