You are given two planes in parametric form, x1 x2 1 x3 where x1, x2, 3,...

 

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You are given two planes in parametric form, x1 x2 1 x3 where x1, x2, 3, ₁, 2, 1,42 € R. Let I be the line of intersection of II and II2. a. Find vectors n₁ and no that are normals to II and II 2 must intersect in a line. b. Find Cartesian equations for II and II₂. II1 x1 0 2 Q-C) -→~~~0 3 x2 + 11 1 +12 2 2 x3 You can make ome checks to a. A possible n₁ is A possible n2 is Πη : you Your v= = 0 3 0 +μ1 2 A + μ2 c. For your first method, assign one of x1, x2 or 3 to be the parameter w and then use your two Cartesian equations for II and II ₂ to express the other two variables in terms of w and hence write down a parametric vector form of the line of intersection L. f. Find m = n₁ x n₂ 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) 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. 3 2 3 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. the right track with your calculations. and II ₂ respectively and explain how you can tell without performing any extra calculations that II 1 3 1 (Use Maple syntax, eg <1, 2, 3>.) (Use Maple syntax, eg <1, 2, 3>.) b. Enter your Cartesian equation for II₁ here: (Use the variable x1, x2 and x3.) (Use the variable x1, x2 and x3.) Enter your Cartesian equation for II here: 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 Maple syntax, eg <1, 2, 3>.) (Use Maple syntax, eg <1, 2, 3>.) You are given two planes in parametric form, x1 x2 1 x3 where x1, x2, 3, ₁, 2, 1,42 € R. Let I be the line of intersection of II and II2. a. Find vectors n₁ and no that are normals to II and II 2 must intersect in a line. b. Find Cartesian equations for II and II₂. II1 x1 0 2 Q-C) -→~~~0 3 x2 + 11 1 +12 2 2 x3 You can make ome checks to a. A possible n₁ is A possible n2 is Πη : you Your v= = 0 3 0 +μ1 2 A + μ2 c. For your first method, assign one of x1, x2 or 3 to be the parameter w and then use your two Cartesian equations for II and II ₂ to express the other two variables in terms of w and hence write down a parametric vector form of the line of intersection L. f. Find m = n₁ x n₂ 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) 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. 3 2 3 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. the right track with your calculations. and II ₂ respectively and explain how you can tell without performing any extra calculations that II 1 3 1 (Use Maple syntax, eg <1, 2, 3>.) (Use Maple syntax, eg <1, 2, 3>.) b. Enter your Cartesian equation for II₁ here: (Use the variable x1, x2 and x3.) (Use the variable x1, x2 and x3.) Enter your Cartesian equation for II here: 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 Maple syntax, eg <1, 2, 3>.) (Use Maple syntax, eg <1, 2, 3>.)

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