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Department of Mathematics MATHS 108 Tutorial 3 – Vectors and Systems of Linear Equations There are three ways of obtaining the marks for this tutorial. Choose any one of them! • Attending a tutorial in person. Attend one of the in-person tutorials and work with your groupmates for a hour on the problems (it’s OK if you don’t solve all of them), sign-in on the sign-in sheet, and you’ll get a mark! If you are using this method, you do not need to upload working to Canvas; just signing in is enough. • Attending an online tutorial. Just like above, but via Zoom. In the Zoom tutorial sessions, you’ll be placed in breakout rooms with other people in the class. To get credit, your group must at some point “Ask for Help” within your breakout room (it’s a button in Zoom that calls the tutor over) to talk with the tutor about your work. When you do this, the tutor will ask for your name/UPI so they can give you a mark. As above, you don’t need to upload any working or solve all of the problems if you’re choosing this method. • Submitting a written document with your answers to Canvas, before 5pm on Friday. If you’re using this method, you need to give an explanation for your answers, including the multiple choice questions. (You don’t need to solve all of them or get them right, but you need to have shown effort on all of them.) If you do this, the markers will give you the mark. Note that markers won’t be able to give feedback on tutorials submitted this way, so if you’d like feedback, please use one of the above two options! Introductions 1. Introduce yourself to everyone in your group. Then, tell everyone about your favourite movie. Past test/exam questions These are problems taken from past Maths 108 tests and exams. Solve them as a group, and try to come to a consensus as to what you believe the answers are! Once your group is agreed, call the tutor over and compare your results with theirs. 1. How many of the following represent a line in R3 ? • x = (2, 1, 3) + s(1, −1, 2) • x = (1, 2, −1) + s(1, 1, −2) + t(2, 2, −4) • x = (1, −1, 2) + s(1, 1, 2) + t(3, −1, 4) • 2x + 3y + z = 2 (a) 3 (b) 2 (c) 1 (d) 4 2. Let P be the plane in R3 given by the general equation x + 2y − 2z = 4 and let Q be the plane in R3 given by the general equation 2x − y − 2z = 3. Which of the following is the angle of intersection of the planes P and Q? MATHS 108 Tutorial 3 – Vectors and Systems of Linear Equations Page 1 of 3 9 −1 5 (b) cos (c) cos (d) cos (a) cos 4 4 2 4 2 3. Which of the following is the row-reduced echelon form of ? 4 6 8 1 2 1 1 0 1 1 0 6 1 0 5 (b) (c) (d) (a) 0 1 −2 0 1 −2 0 1 −2 0 1 1 −1 4 9 −1 4. Given the augmented matrix 4 5 −1 2 4 2 , which of the following is not a valid row operation? 4 6 8 (a) Row 1→ 2×Row 1−2×Row 1 (c) Row 2 → Row 2 −4×Row 1 (b) Swap Row 1 and Row 2 (d) Row 1 → 1 2 Row 1 5. Let P1 , P2 , P3 be three planes in R3 and let A be the augmented matrix corresponding to the three equations of these planes. Suppose that A has reduced row-echelon form: 2 −1 1 2 0 0 0 0 0 0 0 0 Which of the following could be a drawing of the intersections of the three planes P1 , P2 , P3 ? (a) (b) (c) (d) Past assignment questions 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. 2. Let m ∈ R be a constant. Consider the following system of linear equations in the three variables x, y, z: x − 3y + 4z = 6m −2x + my + 4y − 6z − mz = 3 − 12m 4x + my − 14y + 20z = m2 + 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. MATHS 108 Tutorial 3 – Vectors and Systems of Linear Equations Page 3 of 3
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