Review Exercise 1. Determine vector and parametric equations of the plane that contains the points A(1, 2, -1), B(2, 1, 1), and C(3, 1, 4). 2. In question 1, there are a variety of different answers possible, depending on the points and direction vectors chosen. Determine two Cartesian equations for this plane using two different vector equations, and verify that these two equations are identical. 3. a. Determine the vector, parametric, and symmetric equations of the line passing through points A(-3, 2, 8) and B(4, 3, 9). b. Determine the vector and parametric equations of the plane containing the points A(-3, 2, 8), B(4, 3, 9), and C(-2,-1, 3). c. Explain why a symmetric equation cannot exist for a plane. 4. Determine the vector, parametric, and symmetric equations of the line passing through the point A(7, 1, -2) and perpendicular to the plane with equation 2x - 3y + z-1 = 0. 5. Determine the Cartesian equation of each of the following planes: a. through the point P(0, 1, -2), with normal = (-1, 3, 3) b. through the points (3, 0, 1) and (0, 1, -1), and perpendicular to the plane with equation x-y-z+1=0 c. through the points (1, 2, 1) and (2, 1, 4), and parallel to the x-axis 6. Determine the Cartesian equation of the plane that passes through the origin and contains the line = (3, 7, 1) + (2, 2, 3), tER. 7. Find the vector and parametric equations of the plane that is parallel to the yz-plane and contains the point A(-1, 2, 1). 8. Determine the Cartesian equation of the plane that contains the line 7=(2, 3, 2) + t(1, 1, 4), te R, and the point (4, -3, 2). 9. Determine the Cartesian equation of the plane that contains the following lines: L₁:7=(4, 4, 5) + 1(5, -4, 6), te R, and L₂:7 (4,4,5) + s(2, -3, -4), SER 10. Determine an equation for the line that is perpendicular to the plane 3x - 2y + z = 1 passing through (2, 3, -3). Give your answer in vector, parametric, and symmetric form. 11. A plane has 3x + 2yz + 6 = 0 as its Cartesian equation. Determine the vector and parametric equations of this plane. NEL WAW
Review Exercise 1. Determine vector and parametric equations of the plane that contains the points A(1, 2, -1), B(2, 1, 1), and C(3, 1, 4). 2. In question 1, there are a variety of different answers possible, depending on the points and direction vectors chosen. Determine two Cartesian equations for this plane using two different vector equations, and verify that these two equations are identical. 3. a. Determine the vector, parametric, and symmetric equations of the line passing through points A(-3, 2, 8) and B(4, 3, 9). b. Determine the vector and parametric equations of the plane containing the points A(-3, 2, 8), B(4, 3, 9), and C(-2,-1, 3). c. Explain why a symmetric equation cannot exist for a plane. 4. Determine the vector, parametric, and symmetric equations of the line passing through the point A(7, 1, -2) and perpendicular to the plane with equation 2x - 3y + z-1 = 0. 5. Determine the Cartesian equation of each of the following planes: a. through the point P(0, 1, -2), with normal = (-1, 3, 3) b. through the points (3, 0, 1) and (0, 1, -1), and perpendicular to the plane with equation x-y-z+1=0 c. through the points (1, 2, 1) and (2, 1, 4), and parallel to the x-axis 6. Determine the Cartesian equation of the plane that passes through the origin and contains the line = (3, 7, 1) + (2, 2, 3), tER. 7. Find the vector and parametric equations of the plane that is parallel to the yz-plane and contains the point A(-1, 2, 1). 8. Determine the Cartesian equation of the plane that contains the line 7=(2, 3, 2) + t(1, 1, 4), te R, and the point (4, -3, 2). 9. Determine the Cartesian equation of the plane that contains the following lines: L₁:7=(4, 4, 5) + 1(5, -4, 6), te R, and L₂:7 (4,4,5) + s(2, -3, -4), SER 10. Determine an equation for the line that is perpendicular to the plane 3x - 2y + z = 1 passing through (2, 3, -3). Give your answer in vector, parametric, and symmetric form. 11. A plane has 3x + 2yz + 6 = 0 as its Cartesian equation. Determine the vector and parametric equations of this plane. NEL WAW
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter1: Vectors
Section1.3: Lines And Planes
Problem 16EQ
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