MHB Finding the zero vector of a sinusoidal set

Apricity
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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 = {π‘Ž sin (π‘₯+𝑏) | π‘Ž, 𝑏 E R}.

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

𝒔1 +𝒔2 =π‘Ž1 sin (π‘₯+𝑏1)+π‘Ž2 sin (π‘₯+𝑏2)

=(π‘Ž1cosπ‘₯sin𝑏1 +π‘Ž1sinπ‘₯cos𝑏1)+(π‘Ž2cosπ‘₯sin𝑏2 +π‘Ž2sinπ‘₯cos𝑏2)
= (π‘Ž1 sin 𝑏1 + π‘Ž2 sin 𝑏2) cos π‘₯ + (π‘Ž1 cos 𝑏1 + π‘Ž2 cos 𝑏2) sin π‘₯
= π‘Ž3 sin(π‘₯ + 𝑏3),

Where
π‘Ž3 = βˆšπ‘Ž12 + π‘Ž22 + 2π‘Ž1π‘Ž2 cos(𝑏1 βˆ’ 𝑏2), and
tan𝑏3 = π‘Ž1sin𝑏1+π‘Ž2sin𝑏2/π‘Ž1 cos 𝑏1+π‘Ž2 cos 𝑏2

The scalar multiplication is defined as:

πœ†π’”1 = πœ†π‘Ž1 sin(π‘₯ + 𝑏1).

Find the following vectors in 𝑆:
οΏΌοΏΌa. 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!
 
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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.
 
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
 
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