M-273Q Midterm: Montana State MATH 273 exam1Overview2

Find the equation of the plane orthogonal to the line L and passing through P, where L:{x = 4 + t y = 1 - 2t (t epsilon R) z = 8t and P(2, 3, 1) Find the distance between the point Q(2, 0, 1) and the plane pi: -4x + y - z + 5 = 0 Given three nonzero vectors u, v and w in R^n, if the angle between u and w is equal to the angle between v and w, show that w is orthogonal to the vector ||v|| u - ||u||v Use Cauchy-Schwarz inequality to prove that if a_1 and a_2 are nonnegative real numbers, then Squareroot a_1 a_2 lessthanorequalto a_1 + a_2/2.

Does the set (1 + x, 3 - x^2, 1 + 4x + x^2) span P_2? If not, then write one vector as a linear combin other two vectors, then add one of the standard basis vectors of P_2 to the set to form a basis for P_2. Consider the transformation T: R^3 rightarrow R, T(x, y, z) = (x +2y + 3z, y + z, x 3y + 4z, x + z) (a) Find the image of v (2, 3, -5). (b) Find the preimage of w = (0, -1, -1, 2). (c) basis for the range of T (d) Find a basis for the kernel of T.