Advanced Engineering Mathematics
10th Edition
ISBN: 9780470458365
Author: Erwin Kreyszig
Publisher: Wiley, John & Sons, Incorporated
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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.](https://content.bartleby.com/qna-images/question/5290f2ea-32c7-4a56-a195-0d291700f693/dfd87e15-e8c9-4b1f-a566-064235aaebfb/pquh6vn_processed.png)
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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