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Please read the information on Moodle about this assignment. In the question below you will see a faint dividing line. Above that line is the question you should answer in your written work. Below that line you will find some Mobius questions you can use to check some aspects to make sure you are on the right track. Note that the "How did I do?" link is available for this checking. A small part of your mark is assigned to completing this check. Most of your marks are assigned to your written work which must be submitted to Turnitin on Moodle. Do not click the "Submit Assignment" button. Navigate to the last question to ensure your answers are saved. In this question you will find the intersection of two given planes using two different methods. You are given two planes in parametric form, 4 x2 -1 0-0) (²-₂) 3 2 II₁ : + λ1 II₂ : 2 1 -3 +4²1 -2 x1 x2 x3 where #1, #2, #3, 1, 2, M1, M2 R. Let L be the line of intersection of II and II 2. a. Find vectors n₁ and no that are normals to II and II₂ respectively and explain how you can tell without performing any extra calculations that II and II must intersect in a line. b. Find Cartesian equations for the planes II and II₂. 1 (-) 1 3 2 0 2 (²) 3 +12 -2 + 12 c. For your first method, assign one of 1, 2 or 3 to be the parameter w and then use your two Cartesian equations for II and II₂ to express the other two variables in terms of w and hence write down a parametric vector form of the line of intersection L. d. For your second method, substitute expressions for x1, x2 and 3 from the parametric form of II2 into your Cartesian equation for II₁ and hence find a parametric vector form of the line of intersection L. e. If your parametric forms in parts (c) and (d) are different, check that they represent the same line. If your parametric forms in parts (c) and (d) are the same, explain how they could have been different while still describing the same line. f. Find m = n₁ × 12 and show that m is parallel to the line you found in parts (c) and (d). g. Give a geometric explanation of the result in part (f) You can make some checks to ensure you are on the right track with your calculations. a. A possible n₁ is (Use Maple syntax, eg <1, 2, 3>.) (Use Maple syntax, eg <1, 2, 3>.) A possible no is
b. Enter your Cartesian equation for II₁ here: variable x1, x2 and x3.) (Use the Enter your Cartesian equation for II₂ here: variable x1, x2 and x3.) c. A parametric equation of the line will have the form x = a + wv for w E R. Check your values for a and v. Your a = (Use the (Use Maple syntax, eg <1, 2, 3>.) (Use Maple syntax, eg <1, 2, 3>.) Your v = d. You can reuse the answer boxes in the check for part (c) to verify your second parametric form for the line if it is different. e. You're on your own here f. You can check your own m. Remember that the cross product of two vectors is perpendicular to those vectors. g. You're on your own here