If a , ~b , c are three unit vectors such that | a + b |^2+| a + c |^2=8 , then | a +3 ~b |^2+| a +3 c |^2=

Step-by-step solution

Given that a , b , and c are unit vectors, their magnitudes satisfy | a | = | b | = | c | = 1 . The equation | a + b |^2 + | a + c |^2 = 8 expands to: (2 + 2 a b ) + (2 + 2 a c ) = 8 Simplifying yields: 4 + 2( a b + a c ) = 8 Thus a b + a c = 2 The expression | a + 3 b |^2 + | a + 3 c |^2 becomes: (10 + 6 a b ) + (10 + 6 a c ) = 20 + 6( a b + a c ) Substituting the previous result gives: 20 + 6 2 = 32 Final Answer: 32