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2. The line in R³ through the point Po(xo; Yo, 2o) and parallel to the vector v = parametric equations (V1, V2, v3) has x = x0 + tv1, y = Yo + tv2, z = 20 + tv3, -0

2. The line in R³ through the point Po(xo; Yo, 2o) and parallel to the vector v = parametric equations (V1, V2, v3) has x = x0 + tv1, y = Yo + tv2, z = 20 + tv3, -0

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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2. The line in R³ through the point Po(xo, Yo, Zo) and parallel to the vector v = parametric equations (V1, v2, V3) has x = x0 + tv1, y = Yo + tv2, z = zo + tv3, -0 <t <∞. (a) Show that we can rewrite the parametric equations without t as x – xo Y – Yo Z – 20 Vị V2 V3 as long as v1, v2 and vz are nonzero. These are known as the symmetric equations for the line L. (b) We can write the symmetric equations as two distinct equations: x – x0 y – Yo Y – Yo Z – zo and U2 U2 V3 What kind of geometric objects in R³ do these two equations separately represent? (c) Let Li be the line through the point (xo, Yo, 2o) = (1,0, –2) and parallel to v = Following the reasoning of (a) and (b) above, find two planes in R³ such that Lị is their (-1,3, 2). intersection. (d) Let L2 be the line through (xo, Yo, žo) = (1,0, –2) and v = (-1,3,0). Taking note that now v3 = 0, find two planes in R3 such that L2 is their intersection.
Transcribed Image Text:2. The line in R³ through the point Po(xo, Yo, Zo) and parallel to the vector v = parametric equations (V1, v2, V3) has x = x0 + tv1, y = Yo + tv2, z = zo + tv3, -0 <t <∞. (a) Show that we can rewrite the parametric equations without t as x – xo Y – Yo Z – 20 Vị V2 V3 as long as v1, v2 and vz are nonzero. These are known as the symmetric equations for the line L. (b) We can write the symmetric equations as two distinct equations: x – x0 y – Yo Y – Yo Z – zo and U2 U2 V3 What kind of geometric objects in R³ do these two equations separately represent? (c) Let Li be the line through the point (xo, Yo, 2o) = (1,0, –2) and parallel to v = Following the reasoning of (a) and (b) above, find two planes in R³ such that Lị is their (-1,3, 2). intersection. (d) Let L2 be the line through (xo, Yo, žo) = (1,0, –2) and v = (-1,3,0). Taking note that now v3 = 0, find two planes in R3 such that L2 is their intersection.
3. (a) Let P be the plane in R³ defined by the equation n•r = d, where |n = 1. Prove that |d| equals to the distance from the origin to P. (b) Let S be the plane in R3 defined by the equation 2.т — у + 2х — 6. What is the distance from the origin to S? (1, 1, 2) and Q = (2, 1, 2). Note that a plane divides R³ into two half spaces. Do (c) Let P = P, Q lie in the same or in the different half spaces determined by the plane S from (b)?
Transcribed Image Text:3. (a) Let P be the plane in R³ defined by the equation n•r = d, where |n = 1. Prove that |d| equals to the distance from the origin to P. (b) Let S be the plane in R3 defined by the equation 2.т — у + 2х — 6. What is the distance from the origin to S? (1, 1, 2) and Q = (2, 1, 2). Note that a plane divides R³ into two half spaces. Do (c) Let P = P, Q lie in the same or in the different half spaces determined by the plane S from (b)?
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