Linear Algebra: Rotation Matrix Qθ+φ

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Discussion Overview

The discussion revolves around the properties of rotation matrices in linear algebra, specifically focusing on the relationship between consecutive rotations by angles θ and φ, and their combined effect as a single rotation by the angle (θ + φ). The scope includes theoretical exploration and mathematical reasoning.

Discussion Character

  • Exploratory, Mathematical reasoning

Main Points Raised

  • One participant asks for clarification on how to express the combined rotation as a single rotation matrix.
  • Another participant inquires about the specific form of the rotation matrix for an angle θ.
  • A suggestion is made to perform matrix multiplication to investigate the relationship between the two rotations.
  • Repeated inquiry about the rotation matrix with angle θ is noted, along with a reference to relevant trigonometric identities that may assist in the matrix multiplication process.

Areas of Agreement / Disagreement

Participants have not reached a consensus, and multiple viewpoints regarding the approach to the problem remain present.

Contextual Notes

Some participants reference trigonometric identities that are essential for the matrix multiplication, but the discussion does not resolve how these identities apply to the problem at hand.

camchetan
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Show that a rotation by θ followed by a rotation by φ can be expressed as either
two consecutive rotations, or one rotation of (θ + φ). That is, show that Qθ Qφ = Qθ+φ, where Q is the rotation matrix.

Can anyone answer this question I'm a beginner in Linear Algebra
 
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What is the rotation matrix with angle [itex]\theta[/itex]??
 
do the matrix multiplication and see what happens
 
micromass said:
What is the rotation matrix with angle [itex]\theta[/itex]??

genericusrnme said:
do the matrix multiplication and see what happens
You will need the trig identities [itex]cos(\theta+ \phi)= cos(\theta)cos(\phi)- sin(\theta)sin(\phi)[/itex] and [itex]sin(\theta+ \phi)= sin(\theta)cos(\phi)+ cos(\theta)sin(\phi)[/itex]
 

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