10. In precalculus, one of the basic skills you develop is, given a point and a slope, to write down the equation of the corresponding line. In this problem, we will develop the corresponding skill for this course: given a point in R3 and a normal vector, write down the equation of the plane through that point with the given normal vector. This skill will be required through the course. Let p = (P1, P2, P3) in R³, and let N (A, B, C) be the desired normal vector. Let P be the plane we are looking for, that is, through P and with normal vector N. Suppose x = =(x, y, z) is another point on the plane P. Then the vector x - p is parallel to the plane, i.e., if you draw the plane and put the tip of the vector at p, then the whole vector will lie on P. (Draw a picture and convince yourself of this.) = This means that x - p is orthogonal to N, or that N. (x - p) = 0. This is the basic equation of the plane. It will often be written N. x = N. p. Most commonly, it is written Ax+By+Cz = Apı + Bp2 + Cp3. (a) Show that the plane through (1, 1, 1) with normal vector (2, 3, 4) is given by 2x+3y+4z=9. (b) Write the equation of the plane through (1, 2, 3) with normal vector (1, 0, 1). 2
10. In precalculus, one of the basic skills you develop is, given a point and a slope, to write down the equation of the corresponding line. In this problem, we will develop the corresponding skill for this course: given a point in R3 and a normal vector, write down the equation of the plane through that point with the given normal vector. This skill will be required through the course. Let p = (P1, P2, P3) in R³, and let N (A, B, C) be the desired normal vector. Let P be the plane we are looking for, that is, through P and with normal vector N. Suppose x = =(x, y, z) is another point on the plane P. Then the vector x - p is parallel to the plane, i.e., if you draw the plane and put the tip of the vector at p, then the whole vector will lie on P. (Draw a picture and convince yourself of this.) = This means that x - p is orthogonal to N, or that N. (x - p) = 0. This is the basic equation of the plane. It will often be written N. x = N. p. Most commonly, it is written Ax+By+Cz = Apı + Bp2 + Cp3. (a) Show that the plane through (1, 1, 1) with normal vector (2, 3, 4) is given by 2x+3y+4z=9. (b) Write the equation of the plane through (1, 2, 3) with normal vector (1, 0, 1). 2
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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Please show steps thanks. It’s a 2 part question
Transcribed Image Text:10. In precalculus, one of the basic skills you develop is, given a point and
a slope, to write down the equation of the corresponding line. In this
problem, we will develop the corresponding skill for this course: given a
point in R3 and a normal vector, write down the equation of the plane
through that point with the given normal vector. This skill will be required
through the course.
Let p = (P1, P2, P3) in R³, and let N (A, B, C) be the desired normal
vector. Let P be the plane we are looking for, that is, through P and with
normal vector N. Suppose x = (x, y, z) is another point on the plane P.
Then the vector x - p is parallel to the plane, i.e., if you draw the plane
and put the tip of the vector at p, then the whole vector will lie on P.
(Draw a picture and convince yourself of this.)
=
This means that x - p is orthogonal to N, or that N. (x - p) = 0. This
is the basic equation of the plane. It will often be written N. x = N. p.
Most commonly, it is written
Ax+By+Cz = Apı + Bp2 + Cp3.
(a) Show that the plane through (1, 1, 1) with normal vector (2, 3, 4) is
given by 2x+3y+4z=9.
(b) Write the equation of the plane through (1, 2, 3) with normal vector
(1, 0, 1).
2
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