Explain why each set of information does not define a unique plane in three-space. a) x-intercept -3; points P(2, 5, 7) and Q(1, 10, 14) b) points A(12, -3, 6), B(0, -4, 5), and C(-2, 1, 6) c) vector equation [x, y, z] = [5, 1, -2] + s[1, −3, −1] + t[2, −6, −2] d) point P(4, 7, -22) and line [x, y, z]= [1, -2, 5] + t[1, 3, −9] e) parametric equations x = 5-s-2t, y = −2+ 5s + 10t, and z = 9 + 4s + 8t

Give the vector equation of the plane passing through P, Q, and R. P = (8, 8, 8), Q = (4, 0, 2), R = (0, 1, -1) 8 0-0-0- y = + S 8 Z -4 -6 + t

pls help with these thanks. 1. a) Determine a direction vector of the line of intersection of the two planes 6x - 2y + 8z = 14 and 15x - 5y + 20z = 28. If there is not a single line of intersection, classify the system.   Question a) options:   [-7, 3, -5]   [4, -6, 1]   [-6, -5,2]   [1, -7, -4]   parallel and distinct   coincident   b) Solve the following system, if possible. π1:  3x - 2y + 5z = 1 π2:  5x + y - 3z = -4 π3:  6x - 17y + 44z = 31 Find either the point of intersection or the direction vector of the line of intersection or state the system is inconsistent.   Question b) options:   [1, -5, –3]   [-3, 5,-1]   [8, 4, 1]   [-4, -7, 5]   inconsistent

3) Find the general form of the equation of the plane that passes through the points P(-5,4,0), Q(-8,7,-5), and R(-6,9,2). >