Proving Non-Zero Eigenvalues for Rotations in Euclidean Three Space

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In summary, the conversation discusses proving a situation in Euclidean three space where a rotation about the origin, denoted as L, results in L(\vec{v})=\lambda \vec{v}, but neither lambda or the vector v equal zero. The participants consider the concept of a full rotation of 2 pi and the axis of rotation, but express a need for guidance on how to write the proof in mathematical terms.
  • #1
ck22286
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Help! (Euclidean three space)

Homework Statement



Given the Euclidean three space R3and if L is a rotation about the origin, can you prove a situation when L([tex]\vec{v}[/tex])=[tex]\lambda[/tex] [tex]\vec{v}[/tex] and neither lambda or vector v equal zero



Homework Equations





The Attempt at a Solution


I understand that it is a full rotation of 2 pi but I do not know exactly how to prove it.
 
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  • #2


Think about the axis of rotation.
 
  • #3


HallsofIvy said:
Think about the axis of rotation.

I need to see how to make the proof. I understand the concept. I just do not know how to write it down in math terms.
 

What is "Help (Euclidean three space)"?

"Help (Euclidean three space)" is a mathematical concept that refers to a three-dimensional space in which the distance between any two points is calculated using the Euclidean distance formula. It is commonly used in geometry, physics, and engineering.

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The properties of "Help (Euclidean three space)" include having three dimensions (length, width, and height), being continuous and infinite, and following the Euclidean geometry rules of parallel lines, perpendicular lines, and angles.

How is "Help (Euclidean three space)" different from other types of three-dimensional spaces?

"Help (Euclidean three space)" is different from other types of three-dimensional spaces because it follows the Euclidean geometry rules. Other types of three-dimensional spaces, such as non-Euclidean spaces, do not follow these rules and have their own unique properties.

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"Help (Euclidean three space)" has many real-world applications, including in navigation and mapping, computer graphics and animation, physics simulations, and architectural and engineering design.

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"Help (Euclidean three space)" is used in scientific research as a way to model and understand three-dimensional phenomena and relationships. It is particularly useful in fields such as physics, astronomy, and biology.

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