Show that for any vectors a, and b, vector v = a - (a · b / ||b||^2) b is orthogonal to b. It is given: ||m|| = 4; ||n|| = √3; (m, n) = 150°. i. Find the norm of the vector m + 2n. ii. Determine if the vectors (m + 2n) and (- m + n) are orthogonal. Vectors u and v are given by their coordinates in an orthonormal basis {i, j, k}: u = (1, - 3, 3); v = (2, 3, 4). Find: i. the orthogonal vector projection of u onto v; ii. an angle between u and v - u . For each of the following expressions determine whether it is meaningful or meaningless. If it is meaningful, state whether it designates a vector or a scalar: i. (a · b)u ii. u × (a · b) iii. u × (a × b) Evaluate the area of a parallelogram formed by the vectors (a + 2b) and (2a - b), if ||a|| = 1; ||b|| = √2; (a, b) = 135° . Vectors a and b are given by their components in an orthonormal basis {i, j, k}: a = (1, 1, 1); b = (0, 3, 4). Find such a unit vector u that is perpendicular to both vectors a and b , and makes an obtuse angle with the positive Y-semi-axis. Four points are given by their coordinates in a rectangular Cartesian system: M(1, 0, 1); N(0, -1, 3); P(1, 1, 2); and Q(2, 1, 4) . i. Evaluate the area of triangle MNP; ii. Determine, if these four points are coplanar; iii. Evaluate the volume of the tetrahedron MNPQ.