1. Let ā= (0,2,0) and 5 = (0,1, 13). Find: (a) the angle between a and...
Question 1 (2+2+5 marks] (a) Find the angle between the vectors y =(4,0,3), v = (0,2,0). (b) Consider the subspace V (a plane) spanned by the vectors y, V. Find an orthonormal basis for the plane. (Hint: you may not need to use the full Gram-Schmidt process.) (c) Find the projection of the vector w=(1,2,3) onto the subspace Vin (b). Hence find w as a sum of two vectors wi+w, where w, is in V and w, is perpendicular to...
Suppose that ā, 5, and ē are vectors in R2 such that the angles between (a, b) and (ā, 2) are both oblique (i.e., between 0 and 90°). Then the angle between (a, a) where d=5+ Čis: a. Oblique, 0º < 0 < 90° b. Right, 0 = 90° c. Obtuse, 90° <O< 180° d. Straight, 0 = 180° e. Cannot be determined on the basis of the information given.
Number Theory
13 and 14 please!
13)) Let n E N, and let ā, x, y E Zn. Prove that if ā + x = ā + y, then x-y. 14. In this exercise, you will prove that the additive inverse of any element of Z, is unique. (In fact, this is true not only in Z, but in any ring, as we prove in the Appendix on the Student Companion Website.) Let n E N, and let aE Z...
3. Let S be the plane through the points A(0,0,-1), B(0,2,0), C(3,0,0). (a) Find an equation for S of the form ax +by+cz = d. (b) Find the point on that is closest to the point P(2,2,3).
1. Let A= {0,1}2 U... U{0,1}5 and let < be the order on A defined by (s, t) E< if and only if s is a prefix of t. (We consider a word to be a prefix of itself.) (a) Find all minimal elements in A. (Recall that an element x is minimal if there does not exist y E A with y < x.) (b) Are 010 and 01101 comparable? 2. Give an example of a total order on...
The angle between any two vectors can be found from the expression, 7. ā, b = lallbl cos θ Draw the following two vectors on the graph and determine the angle between them a. a=29, b=2+39
Answer each question in the space below. 1. Let A = {0,1} U... U{0,1}5 and let be the order on A defined by (s, t) €< if and only if s is a prefix of t. (We consider a word to be a prefix of itself.) (a) Find all minimal elements in A. (Recall that an element & is minimal if there does not erist Y E A with y < x.) (b) Are 010 and 01101 comparable? 2. Give...
Let →a=2→i−5→j−2→ka→=2i→-5j→-2k→ and →b=5→i−→kb→=5i→-k→. Find
−→a+→b-a→+b→.
Let ā = 27 – 53 – 2k and 7 = 57 - K. Find - ã+ 7. <3i Х 5j k X>
Let θ be an angle in quadrant IV such that cosθ= 12/13
Find the exact values of csc θ and tan θ
Let be an angle in quadrant IV such that cos 0 = 12 13 Find the exact values of csc and tan . 3 csc Х 5 ? tan
Problem l: Let u, v and w be three vectors in R3 (a) Prove that wlv +lvlw bisects the angle between v and w. (b) Consider the projection proj, w of w onto v, and then project this projection on u to get proju (proj, w). Is this necessarily equal to the projection proj, w of w on u? Prove or give a counterexample. (c) Find the volume of the parallelepiped with edges formed by u-(2,5,c), v (1,1,1) and w...