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7=(2, 3, 2) + t(1, 1, 4), tER, and the point (4, -3, 2). of the plane that contains the line 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. Determina OR

7=(2, 3, 2) + t(1, 1, 4), tER, and the point (4, -3, 2). of the plane that contains the line 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. Determina OR

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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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), tER, 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
Transcribed Image Text: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), tER, 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
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