FIGURE 2. A plane E through the origin in Rº with unit normal vector în. For any v e R", we define the reflection of ữ in the plane & by Refle (7) = à – B, %3D where i = Ä+ B and Ä lâî and B || î. Hopefully this coincides with your geometric notion of a reflection in that the vector à in Ɛ is unchanged and the vector B parallel to ân is reversed. The material in Multivariable Calculus I can be used to work out formulas for Ä and B. See Figure 3. Show that Refle (7) = nvn where on the right hand side v and n are viewed as pure quaternions and quaternion multiplication is used.
FIGURE 2. A plane E through the origin in Rº with unit normal vector în. For any v e R", we define the reflection of ữ in the plane & by Refle (7) = à – B, %3D where i = Ä+ B and Ä lâî and B || î. Hopefully this coincides with your geometric notion of a reflection in that the vector à in Ɛ is unchanged and the vector B parallel to ân is reversed. The material in Multivariable Calculus I can be used to work out formulas for Ä and B. See Figure 3. Show that Refle (7) = nvn where on the right hand side v and n are viewed as pure quaternions and quaternion multiplication is used.
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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Hi there, may someone please give me a guide on how to solve this!
Transcribed Image Text:FIGURE 3. Reflection of i in a plane E through the origin.
![First, a minor comment on notation: A vector in space can be
thought of as a pure quaternion. If we consider a vector i = (r, y, z) =
zi + yj + zk E R' as a vector in R' we will write i, or ô when i is a
unit vector. If we consider i as a pure quaternion we will write v € H.
The length of a pure quaternion v is just the length |u] of the vector i.
%3D
Let E be a plane through the origin in R³ with unit normal vector în.
See Figure 2.
FIGURE 2. A plane E through the origin in R' with unit
normal vector în.
For any i E R, we define the reflection of i in the plane E by
Refle (7) = Ä – B,
where i = Ã+ B and A 1 în and B || în. Hopefully this coincides
with your geometric notion of a reflection in that the vector à in E is
unchanged and the vector B parallel to în is reversed. The material in
Multivariable Calculus I can be used to work out formulas for à and B.
See Figure 3.
Show that Refle () = nvn where on the right hand side v and n are
viewed as pure quaternions and quaternion multiplication is used.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F5290f2ea-32c7-4a56-a195-0d291700f693%2Fdfd87e15-e8c9-4b1f-a566-064235aaebfb%2Fpquh6vn_processed.png&w=3840&q=75)
Transcribed Image Text:First, a minor comment on notation: A vector in space can be
thought of as a pure quaternion. If we consider a vector i = (r, y, z) =
zi + yj + zk E R' as a vector in R' we will write i, or ô when i is a
unit vector. If we consider i as a pure quaternion we will write v € H.
The length of a pure quaternion v is just the length |u] of the vector i.
%3D
Let E be a plane through the origin in R³ with unit normal vector în.
See Figure 2.
FIGURE 2. A plane E through the origin in R' with unit
normal vector în.
For any i E R, we define the reflection of i in the plane E by
Refle (7) = Ä – B,
where i = Ã+ B and A 1 în and B || în. Hopefully this coincides
with your geometric notion of a reflection in that the vector à in E is
unchanged and the vector B parallel to în is reversed. The material in
Multivariable Calculus I can be used to work out formulas for à and B.
See Figure 3.
Show that Refle () = nvn where on the right hand side v and n are
viewed as pure quaternions and quaternion multiplication is used.
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