Problem 5. Let H be the plane x + y + z = 0, let în be a unit normal vector to H, and let L be the line chrough the origin which is perpendicular to H. Define two 3 × 3 matrices, as follows: P = îî™ Q = I3 – P a. If i is an arbitrary 3D vector, show that Pi is parallel to L, Qã is parallel to H, and * = Pa + Qã.
Problem 5. Let H be the plane x + y + z = 0, let în be a unit normal vector to H, and let L be the line chrough the origin which is perpendicular to H. Define two 3 × 3 matrices, as follows: P = îî™ Q = I3 – P a. If i is an arbitrary 3D vector, show that Pi is parallel to L, Qã is parallel to H, and * = Pa + Qã.
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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parts a ) - d). For d) can you just compute the matrix F? I can prove that they're orthogonal. Please. I need help.
Transcribed Image Text:Problem 5. Let H be the plane x + y + z = 0, let î be a unit normal vector to H, and let L be the line
through the origin which is perpendicular to H. Define two 3 x 3 matrices, as follows:
P = înT
I A →), and Q = I3 – P
a. If i is an arbitrary 3D vector, show that Pã is parallel to L, Qã is parallel to H,
and
T = Pi + Qã.
b. Let F be the matrix which reflects vectors orthogonally across H. Draw the vectors Pi, Qĩ, and
Fa in the following diagram:
H
c. Give constants a and b such that the matrix
F = aP+ bQ
reflects vectors across H.
d. Compute the matrix F, and verify that it is an orthogonal matrix.
e. Find the matrix of F with respect to the basis B from problem 4.
(Hint: no computation is required, just draw the basis vectors and their transforms in your diagram.)
Note: You may find it enlightening to verify that part a works for any plane through the origin, not just H.
Therefore, this method can be used to compute any 3-dimensional reflection matrix.
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