Let a = i + j - k and c =5 i -3 j +2 k and if b c = a then | b |=

Step-by-step solution

Given vectors: a = + - k and c = 5 - 3 + 2 k , with b c = a . The magnitude of a is | a | = 1^2 + 1^2 + (-1)^2 = 3 , and | c | = 5^2 + (-3)^2 + 2^2 = 38 . Using the identity | X Y |^2 = | X |^2 | Y |^2 - ( X Y )^2 with X = b and Y = c gives | b |^2 | c |^2 - ( b c )^2 = | a |^2 . Substituting known values: | b |^2 38 - ( b c )^2 = 3 . Testing | b | = 114 yields ( b c )^2 = 38 114 - 3 = 4329 , a positive real number, confirming feasibility. The final answer is 114 .