MCV4U d3 - Linear Dependence and Coplanarity Assignment Answer all questions with full solutions. Make sure your work is legible, even after you have scanned it, and submit it as a single file. 1. Write an example of each of the following (assuming it is in 3-dimensional space). a. A point lying on the z-axis. b. A point lying on the xz plane. c. A point lying on both the xy and yz planes. d. A point lying on all three planes. e. A point lying on none of the three planes, but equidistant from the xy and xz planes. 2. Triangle ABC has vertices A(-3, 1, 5), B(-3, 5, 9), and C(-7, 5, 5). What kind of triangle is ABC? Justify your answer. 3. The points (3, -4, 5), (1, 0, 5) and (3, 1, -2) are three of four vertices of parallelogram ABCD. Explain why there are three possibilities for the location of the fourth vertex, and find the three points. 4. The points A(5, -3, z), B(1, 3, 11), and C(x, 15, 27) are collinear. Find the values of x and z. 5. Explain the meaning of direction angles and their relation to direction vectors. a. What are the direction angles of the vector [4, -5, 1]? b. If a point P lies on the y-axis, what are the direction angles of the position vector OP ? 2 2 2 c. Prove that cos cos cos 1 d. A vector has direction angles = 65 and = 45 i. Find the value of ii. Find a vector that has those direction angles e. Explain why it is not possible for two of a vector's direction angles to be less than 45 f. 2 2 2 What is the value of sin sin sin ? Why? 6. Explain the meanings of the terms linearly dependent and coplanar. Make sure you demonstrate that you understand the difference between the terms, and the situation in which linear dependency implies coplanarity. 7. Determine if the vectors [1, -7, 7], [7, 3, -5] and [5, -5, 5] are coplanar. 8. Give examples of sets of three vectors that are a. Collinear b. Coplanar and not collinear c. Not coplanar GO TO: THE DROPBOX AND UPLOAD YOUR WORK. and explain your reasoning. 9. Explain how you would prove if four given points are coplanar. Use your method to determine if A (5, 1, 7), B (13, -5, 17), C (-1, 15, 3) and D (15, 5, 23) are coplanar. 10. Determine if the following vectors are coplanar. Assume that v1 , v2 , and v3 are not coplanar. w1 4v1 3v2 w2 w3 v2 4v3 v1 3v3 GO TO: THE DROPBOX AND UPLOAD YOUR WORK