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Show that the point P(4,-1,1) is common to the lines L, and L, Find The point of intersection of L2 and Lg. A vector parametric equation' of the plane containing the lines L, and La thot

Show that the point P(4,-1,1) is common to the lines L, and L, Find The point of intersection of L2 and Lg. A vector parametric equation' of the plane containing the lines L, and La thot

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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3y+1 Z-2 = 9*1 br-3i- 4j - ¼k + 2(4i + 2j + X-2 13. Given that the lines I, and o where l: - 1 2. -2 3k) a) Find the cosine of the angle between l, and l2 b) Show that l, and l, intersect, giving the position vector of the point of intersection c) Find the vector equation of the line which is perpendicular to l1 and l2 and passes through their point of intersection x+1 4-y Z-2 14. A plane contains the line: and r= (2i+2j+12k) + t(-i+ 2j + 4k). Find: a) 2 2 3 The angle between these lines b) a Cartesian equation of the plane. 15. Find a vector equation of the plane through the points: A(1, 0, 0), B(2, -6, 1) and C(-3, 0 4). Hence, find the coordinates of the point of intersection of the plane ABC and the line L with vector equation: r= 2ti + (1- 5t)j + (t -- 2)k . Find also the sine of the angle between the plane ABC and the line L. 16. Find the sine of the angle which the plane t, given by the equation x + y-z= 24, makes with the line passing through the points with position vectors: i+ 2j + 3k and 4i + 6j + 2k 17. Vector parametric equations of the lines Li and L2 are given by L;:r 2i +j+ (i+j+ 2k), L2:r = 2i +2j + tk + µ(i+ 2j + k), Where t is a constant. Find a) The value of t for which L; and L, intersect b) The position vector of the point of intersection of L, and L2, c) The cosine of the acute angle between L, and L2 d) A vector parametric equation of the plane containing L, and L2. 18. Given the lines X-10 y-1 Z--9 L1: 3 4 L2:r = (-9j + 13k) + u(i + 2j – 3k). Where u is a parameter, x+10 ソ+5 z+4 L3- 4 3 1 a) Show that the point P(4,-1,1) is common to the lines L, and L, | Find b} The point of intersection of L2 and L3. c) A vector parametric equation' of the plane containing the lines L2 and La 19. Given that A is the point (5,-1,2), !I is the plane with vector equation: r. (2i + 6j + 9k) = 311- 33 and O is the origin, find: a) The perpendicular distance of II from 0, b) A vector equation of the linel which passes through .* .d is perpendicular to I, c) The coordinates of the point B where l meetsII. 72
Transcribed Image Text:3y+1 Z-2 = 9*1 br-3i- 4j - ¼k + 2(4i + 2j + X-2 13. Given that the lines I, and o where l: - 1 2. -2 3k) a) Find the cosine of the angle between l, and l2 b) Show that l, and l, intersect, giving the position vector of the point of intersection c) Find the vector equation of the line which is perpendicular to l1 and l2 and passes through their point of intersection x+1 4-y Z-2 14. A plane contains the line: and r= (2i+2j+12k) + t(-i+ 2j + 4k). Find: a) 2 2 3 The angle between these lines b) a Cartesian equation of the plane. 15. Find a vector equation of the plane through the points: A(1, 0, 0), B(2, -6, 1) and C(-3, 0 4). Hence, find the coordinates of the point of intersection of the plane ABC and the line L with vector equation: r= 2ti + (1- 5t)j + (t -- 2)k . Find also the sine of the angle between the plane ABC and the line L. 16. Find the sine of the angle which the plane t, given by the equation x + y-z= 24, makes with the line passing through the points with position vectors: i+ 2j + 3k and 4i + 6j + 2k 17. Vector parametric equations of the lines Li and L2 are given by L;:r 2i +j+ (i+j+ 2k), L2:r = 2i +2j + tk + µ(i+ 2j + k), Where t is a constant. Find a) The value of t for which L; and L, intersect b) The position vector of the point of intersection of L, and L2, c) The cosine of the acute angle between L, and L2 d) A vector parametric equation of the plane containing L, and L2. 18. Given the lines X-10 y-1 Z--9 L1: 3 4 L2:r = (-9j + 13k) + u(i + 2j – 3k). Where u is a parameter, x+10 ソ+5 z+4 L3- 4 3 1 a) Show that the point P(4,-1,1) is common to the lines L, and L, | Find b} The point of intersection of L2 and L3. c) A vector parametric equation' of the plane containing the lines L2 and La 19. Given that A is the point (5,-1,2), !I is the plane with vector equation: r. (2i + 6j + 9k) = 311- 33 and O is the origin, find: a) The perpendicular distance of II from 0, b) A vector equation of the linel which passes through .* .d is perpendicular to I, c) The coordinates of the point B where l meetsII. 72
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