In R2, a rotation fixes a point; in R3, a rotation fixes a line. In R4,what does a rotation fix? Generalize these observations to Rn

Answered: In R2, a rotation fixes a point; in R3,…

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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In R², a rotation fixes a point; in R³, a rotation fixes a line. In R⁴,
what does a rotation fix? Generalize these observations to Rⁿ

Expert Solution

Step 1 Given:

In $R^{2}$ , a rotation fixes a point and in $R^{3}$ , a rotation fixes a line.

The objective is to determine what a rotation fixes in $R^{4}$ then generalize observations to $R^{n}$ .

Step 2

In $R^{2}$ , a rotation is a mapping that takes each point in a figure and rotates it a particular number of degrees around a fixed point ( zero dimension).

In $R^{3}$ , a rotation fixes a line (1 dimensional).

Similarly, In $R^{4}$ , a rotation fixes a plane (2 dimensional).

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