Let a , b , c be three vectors of equal magnitude such that the angle between a and b is , b and c is , c and a is . Then the minimum value of + + is ...

Step-by-step solution

Let the magnitude of the three vectors be | a | = | b | = | c | = x . The square of the magnitude of the sum of the three vectors is given by: | a + b + c |^2 = | a |^2 + | b |^2 + | c |^2 + 2( a b + b c + c a ) Since the square of the magnitude of any vector is always non-negative: | a + b + c |^2 0 | a |^2 + | b |^2 + | c |^2 + 2(| a || b | + | b || c | + | c || a | ) 0 Substituting the magnitudes: x^2 + x^2 + x^2 + 2(x^2 + x^2 + x^2 ) 0 3x^2 + 2x^2( + + ) 0 Since x^2 > 0 (assuming non-zero vectors), dividing the