This is not a project and/or assignment for marks, as theses are textbook questions, and will not be copying the tutors work/answers. Please answer all the questions in the space provided and in full details. Answer in complete sentences. Take as much time as needed to answer all of the questions correct please and thank you. Will write a very neat and wonderful review afterwards. DO NOT COPY AND PASTE OTHER PEOPLES WORK. REFERENCE IS ALREADY PROVIDED TO YOU.

Answer all questions in the space provided. Show ALL your steps in your calculations. Answer in complete sentences when necessary. 1. What is the vector and parametric forms of the equation of the line passing through points P(-2,4) and Q(4, -4)? 2. Four students each made a statement about the equation of a line in R given in vector form, F = (-3,5) + t(2,-3). Which students is/are correct? Justify your answer. i. Eleven: The equation of this line in scalar form is 2x + 3y - 1 = 0 ii. Lucas: A line L, that is parallel to the given line is as follows: .S x = 1 + 2.5t ly =8-3.75t 111. Finn: A vector that is perpendicular to the given line is D = (9, -4) iv. Dustin: Two points on this line are: (-1,2) and (0,0.5) 3. A point P(3,0,5) lies on a plane that can be defined by the scalar equation Ax + By + Cz + D = 0. A normal vector, n. to this plane is (2, -3,1). Find the scalar equation. 4. How many solutions are there between the plane P, : (x, y, z) = (4,-3, -1) + s(1, -3,1) + t(2,4,-3) and the line Ly: (x, v,z, ) = (3,1 -2) + k(-1, -1,1)? If there is/are solutions, find the value of the solution(s).5. How many solutions are there between planes P1 :2x - y + 2z = 2 and P2: - x + 2y + z = 1? If there is/are solutions, find the value of the solution(s). 6. Dustin is trying to design a weapon to fight the Demogorgon. His model has three planes, and he needs to figure out their configuration. He thinks they intersect in a line or at a point. Is he correct? Justify your answer. If there is/are solutions, find the value of the solution(s) Pix + 4y+32 -5 =0 P2:x + 3y+2z -4 =0 Pa:xty- z+1=0 7. Lucas does not like Dustin weapon and decides to design his own. His model has a plane that contains the point A(-5,2, -4) and two direction vectors, AB = (4,8,1)and AC = (-3, -2,0). Which of the following statement(s) about this plane is/are true? Justify your answer. i. The plane has a parametric equation that can be defined as follows: (x = -5+4t - 3s y = 2+8t - 2s z =-4+t 1i. The plane has a vector equation that can be defined as (x, y, z) = (-5,2, -4) + t(3,2,0) + s( 4,8,1 ) 111. The point D (31,6,0) lies on the plane iv. The z-intercept of the plane is 0,0, -5)