Finding the zero vector of a sinusoidal set

[Apricity]

Hi everyone,

Thank you in advance for taking the time to read my question and for your help. I really appreciate it.

The question is set out as follows:

Given the set

𝑆 = {𝑎 sin (𝑥+𝑏) | 𝑎, 𝑏 E R}.

The addition of elements 𝒔1, 𝒔2 ∈ 𝑆 is defined as follows:

𝒔1 +𝒔2 =𝑎1 sin (𝑥+𝑏1)+𝑎2 sin (𝑥+𝑏2)

=(𝑎1cos𝑥sin𝑏1 +𝑎1sin𝑥cos𝑏1)+(𝑎2cos𝑥sin𝑏2 +𝑎2sin𝑥cos𝑏2)
= (𝑎1 sin 𝑏1 + 𝑎2 sin 𝑏2) cos 𝑥 + (𝑎1 cos 𝑏1 + 𝑎2 cos 𝑏2) sin 𝑥
= 𝑎3 sin(𝑥 + 𝑏3),

Where
𝑎3 = √𝑎12 + 𝑎22 + 2𝑎1𝑎2 cos(𝑏1 − 𝑏2), and
tan𝑏3 = 𝑎1sin𝑏1+𝑎2sin𝑏2/𝑎1 cos 𝑏1+𝑎2 cos 𝑏2

The scalar multiplication is defined as:

𝜆𝒔1 = 𝜆𝑎1 sin(𝑥 + 𝑏1).

Find the following vectors in 𝑆:
a. The zero vector
b. The negative of
c. The vector
d. The vector
e. The vectorI understand that these are phasors with the same frequency and different amplitude and phase. I also understand the definition of the zero vector which is:

Given a vector u, and a vector v,
u+v=u=v+u

I just don't know where to go from here to find the requested vectors. I have researched online for two days and I can't seem to understand the process. Please help me!

Thank you!

 

[HallsofIvy]

functions of the form "a sin(x+ b)" are a subset of the set of "all functions" and the addition you describe is the "usual" addition of functions- that is, this is a subspace of the space of all functions and so has the same "zero vector", the "zero function", f(x)= 0 for all x. Specifically that is the function a sin(x+ b) with a= 0.

 

[Apricity]

Thanks for your reply HallsofIvy..That was my conclusion too... I thought the only way a sinusoidal wave can be zero is if it's amplitude is zero, but somehow I was complicating myself unnecessarily... It just seemed too simple ...

Thanks again