Let a = 2 i + 3 j + 3 k and b = 6 i + 3 j + 3 k . Then the square of the area of the triangle with adjacent sides determined by the vectors (2 a + 3 b ) and ( a - b ) is :

Step-by-step solution

The area of a triangle with adjacent sides u and v is given by = 1 2 | u v | . Here, u = 2 a + 3 b and v = a - b . u v = (2 a + 3 b ) ( a - b ) u v = 2( a a ) - 2( a b ) + 3( b a ) - 3( b b ) Since a a = 0 , b b = 0 , and b a = - a b , we get: u v = -2( a b ) - 3( a b ) = -5( a b ) The area of the triangle is = 1 2 |-5( a b )| = 5 2 | a b | . Now, calculating a b : a b = vmatrix i & j & k 2 & 3 & 3 6 & 3 & 3 vmatrix a b = i (9 - 9) - j (6 - 18) + k (6 - 18) = 12 j - 12 k The square of the magnitude is: | a b |^2 =