If the projection of a on b + c is twice the projection of b + c on a also if | b |=2 2 ,| c |=4 and the angle between b and c is 4 then | a |=

Step-by-step solution

The projection of a on ( b + c ) is twice that of ( b + c ) on a , yielding the relation: a ( b + c ) | b + c | = 2 a ( b + c ) | a | Canceling the common nonzero dot product and simplifying gives: | a | = 2 | b + c | The magnitude of b + c is computed using the given data: | b | = 2 2 , | c | = 4 , and = 4 . | b + c |^2 = | b |^2 + | c |^2 + 2 b c = (2 2 )^2 + 4^2 + 2 (2 2 4 4 ) | b + c |^2 = 8 + 16 + 2 (8 2 1 2 ) = 24 + 16 = 40 Thus | b + c | = 40 = 2 10 , and substituting into the earlier relation gives: | a | =