W7 - Intersection of Planes

W7-Intersection of Planes Unit 6 MCV4U Jensen SOLUTIONS Determine the solution(s) to each system of planes. If there are no consistent solutions, give evidence and or an explanation of why. a) pi_{1} / 3 * x - 2y + z = 4 pi_{2} / 6 * x - 4y + 3z = 7 a) normals are not parallel; of Intersect in a line. b) pi_{1} / 2 * x - 8y - 6z - 2 = 0 pi_{2} / (- x) + 4y + 3z - 5 = 0 6x - 4y + 2z =-6x - 4y + 3z = 7 2 * 0 z = - 1 - z = 1 c) pi_{1} / x - 2y + 2.5z - 1 = 0 pi_{2} / 3 * x - 6y + 7.5z - 3 = 0 d) pi_{1} / x + y + 2z = 5 pi_{2} / 4 * x - 3y + z = - 8 pi_{3} / - 5 * x - 2y + 3z = 7 e) pi_{1} / x + y + z = - 3 pi_{2} / 2 * x + 2y - 3z = 4 pi_{3} / 3 * x + 3y - 2z = 1 Let x = t 3t - y = 2y 3t - 2y + (- 1) = 4 3/2 * t - 5/2 = y 3t - 5 = 2y [x, yzz] =[0,-2,5,-1]+t[i,1.5, ] f = x = t; y = 3/8 * t - 5/2; z = - 1 pi_{2} / 2 * x - 2y - 6z = - 4 pi_{3} / - 3 * x + 3y + 9z = 6 pi_{1} / (- x) + y + 3z = 2 g) pi_{1} / x - 4y - 13z = 4 pi_{3} / - 3 * x + 5y + 4z = 2 pi_{2} / x - 2y - 3z = 2 h) pi_{1} / 3 * x - 2y + 5z = 1 pi_{2} / 5 * x + y - 3z = - 4 pi_{3} / x - 5y + 13z = 6 ⅰ) pi_{1} / (- x) + y + 3z = 2 pi_{2} / (- x) + y + 3z = 4 pi_{3} / x - 3y + 5z = 6 j) pi_{1} / x + 4y + 3z = 5 pi_{2} / x + 3y + 2z + 4 = 0 pi_{3} / x + y - z = - 1 b) normal ARE parallel; check it coincident of disting. 2x - 8y - 6z - 2 = 0 2x + 8y + 6z - 10 = 0 + -12:0 o No solutiens Parallel and distinct. normals are parallel; check if distinct of coincident 3X 3x - 6y + 7.5z - 3 = 0 ② 3x - 6y + 7.5z - 3 = 0 0:0 -oc parallel and coincident & infinite solutions. d) no normals are parallel x + y + 2z = 5 8x - 6y + 2z = - 16 - 7x + 7y = 21 3x 3x + 3y + 6z =|5 2x - 6x - 4y + 6z = 14 13x + 7y = 1 137x+74=1-- 7x + 7y = 21 - 20yx = 20 x = - 1 13(-1) 7y = 1 13x + 7y = 1 7y = 14 y = 2 x + y + 2z = 5 - 1 + 2 + 2z = 5 z = 2 2z = y The intersect at the point (-1,2,2) e) 2x 2x + 2y + 2z = - 6 2x + 2y - 3z = 4 5z = - 10 z = - 2 3x 3x + 3y + 3z = - 9 3x + 3y - 2z = 1 5z = - 10 z = - 2 - let x = t t + y + (- 2) = - 3 x + y + z = - 3 y = - 1 - t [x, y, z] =[0,-1,-2]+t[|,-1,0] l = x + t; y = - i - t; z = - 2 3pi_{1} = pi_{3} Three coincident planes. Infinite solutions. - 2pi_{1} = T_{2} (6 No parallel normal s x - 4y - 13z = 4 x - 2y - 3z = 2 - 2y - oz = 2 3x - 12y - 39z = 12 ③ - 3x + 5y + 4z = 2 - 7y - 35z = 14 7(0 - 6) ) - 14y - 7oz = 14 2x (3x+3) -144-707=28 -0= -14 No. Solutions. It is an inconsistent systen. h) 540 15x - 10y + 25z = 5 3 15x + 3y - 9z = - 12 ④ - 13y + 3 / z = 17 3x - 2y + 5z = 1 33 3x - 15y + 39z = 18 13y - 34z = - 17 - 13y + 34z = 17 13y - 34z = - 17 00 & they intersect in a live. Let yot -13€ +347=17 34z = 17 + 13t z = 17/34 + 13/34 * t x - 5y + 13z = 6 kappa - 5t + 13(17/34 + 13/34 * t) = 6 [x, y, z] =[ matrix - 1 2 ,0, 17 34 matrix ]+t[ 1 34 ,1, 13 34 matrix ] x - 5t + 221/34 + 169/34 * t = 6 chi = - 1/2 + i/(3H) * t underline f = int matrix x= 1 2 + 1 34 t\\ y=E\\ z= 7 34 + 13/34 * t matrix ⅰ) vec n_{1} = vec n 2 but pi 1 ne k*pi_{2} ; de paraulel and distinct. f an inconsistent systen. J) on back. Answers: a) [x, y, z] = [5/3, 0, - 1] + t[2/3, 1, 0] b) c) Infinite solutions, parallel and coincident d) (-1,2,2) e) [x, y, z] = [- 1, 0, - 2] + s[- 1, 1, 0] f) planes g) inconsistent system; the planes intersect in pairs [x, y, z] = [- 7/13, - 17/13, 0] + t[1/13, 34/13, 1] distinct planes intersected by a third plane (-52,30)-21) j) x + 4y + 3z = 5 x + 3y + 2z = - 4 ( yy + z = 9 x + 4y + 3z = 5 x + y - z = - 1 3y + 4z = 6 344 3y + 3z = 27 ⑤ 3y + 4z = 6 - z = 21 z = - 21 & intersect at a point y + (- 21) = 9 y = 30 x + 4(30) + 3(- 21) = S x = - 52 They Intersect of (-52, 30,-21) 10:6+35 0-2-t 3-22) 11310107 (0,0,1)