Linear Algebra: Rotation Matrix Qθ+φ

[camchetan]

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

 

[micromass]

What is the rotation matrix with angle $\theta$??

 

[genericusrnme]

do the matrix multiplication and see what happens

 

[HallsofIvy]

What is the rotation matrix with angle $\theta$??

do the matrix multiplication and see what happens

You will need the trig identities $cos(\theta+ \phi)= cos(\theta)cos(\phi)- sin(\theta)sin(\phi)$ and $sin(\theta+ \phi)= sin(\theta)cos(\phi)+ cos(\theta)sin(\phi)$

 

[blue_raver22]

Sure, I can help you with this question. Let's start by defining the rotation matrix Qθ as follows:

Qθ = [cosθ -sinθ
sinθ cosθ]

Similarly, the rotation matrix Qφ can be defined as:

Qφ = [cosφ -sinφ
sinφ cosφ]

Now, let's consider the product Qθ Qφ, which represents a rotation by θ followed by a rotation by φ. We can express this product as follows:

Qθ Qφ = [cosθ -sinθ
sinθ cosθ] [cosφ -sinφ
sinφ cosφ]

Using matrix multiplication, we get:

Qθ Qφ = [cosθ cosφ - sinθ sinφ -cosθ sinφ -sinθ cosφ
sinθ cosφ + cosθ sinφ -sinθ sinφ + cosθ cosφ]

Simplifying this expression, we get:

Qθ Qφ = [cos(θ+φ) -sin(θ+φ)
sin(θ+φ) cos(θ+φ)]

This is exactly the rotation matrix for the rotation by (θ+φ). Therefore, we have shown that Qθ Qφ = Qθ+φ, which means that a rotation by θ followed by a rotation by φ can be expressed as one rotation of (θ+φ).

Alternatively, we can also express the product Qθ Qφ as two consecutive rotations. First, we rotate by θ and then by φ. This can be written as:

Qθ Qφ = [cosθ -sinθ
sinθ cosθ] [cosφ -sinφ
sinφ cosφ]

= [cosθ cosφ - sinθ sinφ -cosθ sinφ -sinθ cosφ]
[sinθ cosφ + cosθ sinφ -sinθ sinφ + cosθ cosφ]

= [cosθ cosφ - sinθ sinφ sinθ cosφ + cosθ sinφ
sinθ cosφ + cosθ sinφ -sinθ sinφ + cosθ cosφ]

= [cosθ cosφ - sinθ sinφ sinθ cosφ + cosθ sinφ
sinθ cosφ + cosθ sinφ -sinθ sinφ + cosθ cosφ]

= [cosθ cosφ - sinθ sinφ cos(π/2 - φ) + sin(π/2 - θ) sinφ