Let a = i + 2 j + k and b = i + j + k be two vectors, where , Z . If the vectors ( a + b ) and ( a - b ) are mutually perpendicular, and the area of the parallelogram formed by a and b is 65 , then the value of ^2 + ^2 is equal to

Step-by-step solution

Since ( a + b ) and ( a - b ) are mutually perpendicular, their dot product is zero: ( a + b ) ( a - b ) = 0 | a |^2 - | b |^2 = 0 | a |^2 = | b |^2 Substituting the components of a and b : ^2 + 2^2 + 1^2 = 1^2 + ^2 + ^2 ^2 + 5 = ^2 + ^2 + 1 ^2 = 4 Since Z , = 2 . The area of the parallelogram formed by a and b is given by | a b | . We are given | a b | = 65 , so | a b |^2 = 65 . Calculating the cross product: a b = vmatrix i & j & k & 2 & 1 1 & & vmatrix = (2 - ) i - ( - 1) j + ( ^2 - 2) k The squared magnitude is