Let u and v be two vectors such that for all real numbers x and y , the relation |x u + y v |^2 = 14x^2 + 6y^2 - 4xy holds true. The area of the triangle with adjacent sides u - v and u + 2 v is equal to

Step-by-step solution

Given that |x u + y v |^2 = 14x^2 + 6y^2 - 4xy for all x, y R . Expanding the left side, we get: x^2| u |^2 + y^2| v |^2 + 2xy( u v ) = 14x^2 + 6y^2 - 4xy Since this is an identity in x and y , we can equate the corresponding coefficients: | u |^2 = 14 | v |^2 = 6 2( u v ) = -4 u v = -2 Using Lagrange's identity, we find the square of the magnitude of the cross product: | u v |^2 = | u |^2| v |^2 - ( u v )^2 | u v |^2 = (14)(6) - (-2)^2 = 84 - 4 = 80 | u v | = 80 = 4 5 The area of the triangle with adjacent sides u