Answer Happy

In this question you will find the intersection of two planes using two different methods. You are given two planes in parametric form, II₁ : II₂ : 3 #1 Q-C ~C 12 0 + A1 0 -3 13 Q-0-0-0 +41 12 2 2 -1 2 13 where #1, #2, #3,1,2,1,42 € R. Let I be the line of intersection of II and II₂. a. Find vectors ₁ and ₂ that are normals to II and II must intersect in a line. +442 and II respectively and explain how you can tell without performing any extra calculations that II ₁ b. Find Cartesian equations for II and II ₂. 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 II 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 2₁, 2 and 3 from the parametric form of II into your Cartesian equation for II 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. QI f. Find m = n₁ x n 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 m₁ is A possible na is b. Enter your Cartesian equation for II here: Enter your Cartesian equation for II ₂ here: c. A parametric equation of the line will have the form x = a + wv for w€ R. Check your values for a and v. Your a = (Use Maple syntax, eg <1, 2, 3>.) (Use Maple syntax, eg <1, 2, 3>.) (Use Maple syntax, eg <1, 2, 3>.) (Use the variable x1, x2 and x3.) (Use the variable x1, x2 and x3.) Your v = (Use Maple syntax, eg <1, 2, 3>.) 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