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72 Find the shortest distance between the two lines r=(4,-2, 3) + (2, 1, -1) and r= = (-7, -2, 1) + s(3, 2, 1)

72 Find the shortest distance between the two lines r=(4,-2, 3) + (2, 1, -1) and r= = (-7, -2, 1) + s(3, 2, 1)

Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter1: Vectors
Section1.3: Lines And Planes
Problem 18EQ
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72 Find the shortest distance between the two lines r=(4,-2, 3) + (2, 1, -1) and r= (-7, -2, 1) + s(3, 2, 1)
Transcribed Image Text:72 Find the shortest distance between the two lines r=(4,-2, 3) + (2, 1, -1) and r= (-7, -2, 1) + s(3, 2, 1)
70 P is a point on a straight line with position vector r = a + tb. Show that 7=a+2a-bt+ B² By completing the square, show that is a minimum for the point P for which t-a-b/B. Show that at this point OP is perpendicular to the line r= a + tb. (This proves the well-known result that the shortest distance from a point to a line is the length of the perpendicular from that point to the line.)
Transcribed Image Text:70 P is a point on a straight line with position vector r = a + tb. Show that 7=a+2a-bt+ B² By completing the square, show that is a minimum for the point P for which t-a-b/B. Show that at this point OP is perpendicular to the line r= a + tb. (This proves the well-known result that the shortest distance from a point to a line is the length of the perpendicular from that point to the line.)
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