Passage: Let a , b , c , a + b , b + c , a + b + c be unit vectors. Question: What is the angle between a and c ?

Step-by-step solution

Given | a | = | b | = | c | = 1 | a + b | = 1 | a |^2 + | b |^2 + 2 a b = 1 1 + 1 + 2 a b = 1 a b = - 1 2 | b + c | = 1 | b |^2 + | c |^2 + 2 b c = 1 1 + 1 + 2 b c = 1 b c = - 1 2 | a + b + c | = 1 | a |^2 + | b |^2 + | c |^2 + 2( a b + b c + c a ) = 1 1 + 1 + 1 + 2 (- 1 2 - 1 2 + c a ) = 1 3 + 2(-1 + c a ) = 1 1 + 2 c a = 1 c a = 0 Since the dot product of a and c is 0 , the angle between them is 2 . Answer: 2