You are given two planes in parametric form,x1Il : x2X1IL : x2=|0 +41 2 + 2 4X3whereX1 , X2 , X3, 11 , d2,41, H2 E R. LetL be the line of intersection ofп andIL.a. Find vectorsnj andnɔ that are normals toП andIL respectively and explain how you can tell without performing any extra calculations thatII andII, must intersect in a line.b. Find Cartesian equations forП andIL.c. For your first method, assign one ofX2 orX3 to be the parametero and then use your two Cartesian equations forП andII, to express the other two variables in terms ofo and hence write down a parametric vector form of the line of intersectionL.d. For your second method, substitute expressions forX1 »X andX3 from the parametric form ofII, into your Cartesian equation forII, and hence find a parametric vector form of the line of intersectionL.e. If your parametric forms in parts (c) and (d) are different, check that they represent the same line. If your parametric forms in parts (c) and (d) are the same, explain how they could havebeen different while still describing the same line.f. Findm = nj x n, and show thatm is parallel to the line you found in parts (c) and (d).g. Give a geometric explanation of the result in part (f)

The planes are given by π1: x1x2x3=0-33+λ11-12+λ2313 and π2: x1x2x3=000+μ112-1+μ2340. For π1:…

a. Find vectors nį and n2 that are normals to II and IL respectively and explain how you can tell without performing any extra calculations that Ilj and IL must intersect in a line. b. Find Cartesian equations for Пр and c. For your first method, assign one of *2 or x3 to be the parameter o and then use your two Cartesian equations for п аnd IL to express the other two variables in terms of o and hence write down a parametric vector form of the line of intersection L.

a. Find vectors nį and n2 that are normals to II and IL respectively and explain how you can tell without performing any extra calculations that Ilj and IL must intersect in a line. b. Find Cartesian equations for Пр and c. For your first method, assign one of *2 or x3 to be the parameter o and then use your two Cartesian equations for п аnd IL to express the other two variables in terms of o and hence write down a parametric vector form of the line of intersection L.

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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Question

You are given two planes in parametric form,
x1
Il : x2
X1
IL : x2=|0 +41 2 + 2 4
X3
where
X1 , X2 , X3, 11 , d2,41, H2 E R. Let
L be the line of intersection of
п and
IL.
a. Find vectors
nj and
nɔ that are normals to
П and
IL respectively and explain how you can tell without performing any extra calculations that
II and
II, must intersect in a line.
b. Find Cartesian equations for
П and
IL.
c. For your first method, assign one of
X2 or
X3 to be the parameter
o and then use your two Cartesian equations for
П and
II, to express the other two variables in terms of
o and hence write down a parametric vector form of the line of intersection
L.
d. For your second method, substitute expressions for
X1 »
X and
X3 from the parametric form of
II, into your Cartesian equation for
II, and hence find a parametric vector form of the line of intersection
L.
e. If your parametric forms in parts (c) and (d) are different, check that they represent the same line. If your parametric forms in parts (c) and (d) are the same, explain how they could have
been different while still describing the same line.
f. Find
m = nj x n, and show that
m is parallel to the line you found in parts (c) and (d).
g. Give a geometric explanation of the result in part (f)

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