Let (x^2 + y^2 = 1 ) (u^2 + v^2 = 1 and ) (xu + yv = 0, then) 1. (x^2 + u^2 = 1) 2. (y^2 + v^2 = 1 ) 3. (xy + uv = 0) Which of the above is/are true?

The given equations define two vectors, A = (x, y) and B = (u, v), in R^2. The conditions x^2 + y^2 = 1 and u^2 + v^2 = 1 indicate that both are unit vectors. The condition xu + yv = 0 signifies that their dot product is zero, meaning they are orthogonal [1]. In a 2D space, if two unit vectors are orthogonal, they form an orthonormal basis. This implies that the matrix M = [[x, u], [y, v]] is an orthogonal matrix. For any orthogonal matrix, the rows must also be orthonormal. The first row (x, u) being a unit vector implies x^2 + u^2 = 1 (Statement 1). The second row (y, v) being a unit vector implies y^2 + v^2 = 1 (Statement 2). Finally, the dot product of the rows must be zero, leading to xy + uv = 0 (Statement 3). Thus, all three statements are mathematically true.

Sources

1. [1] https://www.ucl.ac.uk/~ucahmdl/LessonPlans/Lesson10.pdf