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MCV4U EQUATIONS OF PLANES IN R3 PRACTICE 1) Does each point lie on the plane 3x-5y-6z = 12 a) A(0, 0, -2) b) B(1, 1, -2) 2) Find the x-, y-, and z-intercepts of each plane. a) 2x - 3y + 6z = 12 b) 4x - 8z = 16 3) Write the parametric equations of each plane given its vector equation. a) [x, y, z] = [3, -1,2] + t[4, -2,3] + s[5,1.0] b) [x, y, z] = [0,8,7] + t[1,0, -3] + s[1, -4.7] 4) Write the vector equation of a plane given its parametric equations. x = 1-9t + 4s ( x = 4 -t -s a)my=8-7t +s b) m: y = 3+4s =-1-3t + 2s z = 2t 5) Determine the x-, y-, and z-intercepts of the plane [x, y. z] = [1,-3,2] + +[4, - 3.5] + s[-1,7,0] 6) Write a vector equation for each plane: a) contains the origin; has direction vectors a = [2, -1, 7] and b = [3, 5, 2] b) contains the points D(1, -2, 3), E(5, -1, 8), and F(3, 9, 2) c) contains the point Po(2, -1, 5); parallel to the xy-plane x =7+3t d) has y-intercept -7; parallel to the plane defined by the parametric equations y = 6 + 2t - 5s (= =1-8t + 3s 7) Determine the vector equation of the plane that contains the points A(2, -1, 4), B(-3, 4, 5), and C(8, -1, 6). 8) A plane is perpendicular to [x, y, z] = [1, -10,8] + s[1,2, -1] and contains the point P(-1,4, -2). Determine if the point A(7,10,16) is also on this plane. Answers: 1)a) Yes b) No 2)a) x-int: (6, 0, 0); y-int: (0, -4, 0); z-int: (0, 0, 2) b) x-int: (4, 0, 0); y-int: none; z-int: (0, 0, -2) 3)a) m. ( x = 3 + 4t + 5s y=-1-2t+s z=2+3t b) m: (x=t+s y=8 -4s z=7 - 3t +7s 4)a) [x, y, z] = [1, 8, -1] + [[-9, -7, 3] + s[4, 1, 2] b) [x, y, z] = [4, 3, 0] + /[-1, 0, 2] + s[-1, 4, 0] 5) x-int: (- 6/7, 0, 0); y-int: (0, -6, 0); z-int: (0, 0, 6/5) 6) a) [x, y, z] = [0, 0, 0] + ([2, -1, 7] + s[3, 5, 2] b) [x, y, z] = [1, -2, 3] + f[4, 1, 5] + s[2, 11, -1] c) [x, y, z] = [2, -1, 5] + +[1, 0, 0] + s[0, 1, 0] d) [x, y, z] = [0, -7, 0] + ([3, 2, -8] + s[0, -5, 3] 7) [x, y, z] = [2, -1, 4] + ([-5, 5, 1] + s[6, 0, 2] 8) Not on the plane