ME641 Engineering Analysis I Homework Assignment 1 ME641 Homework Assignment 1 Problem 1 Show that if x is a nonzero vector, is a unit vector. Problem 2 Show that the oblique rank 2 projection of vector x along vector a onto a planar surface with normal b is a projection (i.e. the second projection does not change the image of the first projection). Problem 3 Consider the following vectors expressed in Cartesian coordinates: a) Determine whether vectors x1, x2 and x3 are linearly independent. b) Determine whether vectors x1, x2 and x3 are orthogonal. c) Determine whether vector a can be expressed as a linear combination of vectors x1, x2 and x3 and briefly explain why or why not. If so, find its components a1, a2 and a3. Problem 4 Show that the magnitude of the cross product ab is the area of the parallelogram determined by a and b. Problem 5 Determine whether vectors x and y are orthogonal. Problem 6 Show which of the following subsets of R3 are subspaces and which are not: a) The set of vectors in R3 with the first components 1. b) All vectors with x3 = 0. c) The set of vectors with nonzero first component. -1/2- 2017 Sven K. Esche ME641 Engineering Analysis I Homework Assignment 1 Problem 7 Determine whether the following vectors are linearly independent: a) [1,2,1,2] [ 2,5,3,0] [1,0,1,10] b) [3,0,0,2] [1,0,0,4] Problem 8 The vector space of second-order polynomials on the interval 1 x 1 is spanned by the following vectors: 0 (x ) = 1 1 (x ) = x 2 (x ) = x 2 a) Show that these vectors are not orthogonal. b) Using the Gram-Schmidt process, find an orthonormal basis for this vector space. Problem 9 Show that the Legendre polynomials P0 (x ) = 1 P1 (x ) = x P2 (x ) = the interval 1 x 1 . -2/2- 3 2 1 x are orthogonal on 2 2 2017 Sven K. Esche