3. Which of the following is the row-reduced echelon form of 4 4 6 8 (a) [6 1 2] () [87 2] () [81 2] ([618] 4. Given the augmented matrix 2 4 2 8 which of the following is not a valid row operation? (a) Row 1-+ 2xRow 1-2xRow 1 (c) Row 2 -+ Row 2 -4xRow 1 (b) Swap Row 1 and Row 2 (d) Row 1 - - Row 1 5. Let P1, P2, P3 be three planes in RS and let A be the augmented matrix corresponding to the three equations of these planes. Suppose that A has reduced row-echelon form: 0 Which of the following could be a drawing of the intersections of the three planes P1, P2, P3? (a) (b) (c) (d)1. Hos;r many of the following represent a line in R3? 1- 3:: [2,1,3] +e(1,1,2] e x = [1,2, 1) + 3(1, 1, 2] +t{2,2, 4]I o x = [1, 1, 2) + 3(1, 1, 2] + t[3, 1,4) l 2.: + 3y+ z = 2 {a} 3 {b} 2 {c} 1 {d} 4 2. Let P be the plane in R3 given by the general equation a: + 2;.- 2: = 4 and let (.2 be the plane in R3 given by the general equation 25'; e y 22: = 3. Which of the following is the angle of intersection of the planes P and Q? MATHS 111-3 Tutorial 3 - Vectors and Systems of linear Equations Page 1 of 3 2. Let m E R be a constant. Consider the following system of linear equations in the three variables T, V, Z: x - 3y + 42 = 6m -2x + my + 4y - 6z - mz = 3 - 12m 4x + my - 14y + 20z = m' + 23m - 3 (a) Put the augmented matrix corresponding to this system of equations into row echelon form. Note that your answer will likely have several cases depending on the value of m. (b) Determine the value(s) of m that will give: i. A unique solution. ii. An infinite number of solutions. iii. No solutions.These are problems taken from past assignments. We're not expecting most people attending the tutorials to get through these! Instead, they're here so that you can get practice with long-form questions. Have fun, and do try all of them before checking the solutions! If you are submitting your answers to Canvas to get the mark, you must attempt all these questions as well. 1. (a) What geometric object does the equation x = 0 correspond to in i. 1 dimension? ii. 2 dimensions? iii. 3 dimensions? (b) Consider the three points A(2,3, 1), B(-3, 2, 2) and C(1, -1,3), as well as the plane P defined by x - 2y + 3z = -4. i. Find the vector equation of the line through A and B. ii. Show that this line does not intersect the plane P. iii. Find the vector equation of the line through A and C. Find its intersection with P. MATHS 108 Tutorial 3 - Vectors and Systems of Linear Equations Page 2 of 3 iv. Find the vector equation of a line passing through the point B, not passing through the point A, and which does not intersect the plane P. Hint: Try finding a plane Q which is parallel to P and contains B. Then choose a line from within this plane that does not pass through A
