Vector valued velocity and acceleration question

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When integrating vector-valued functions from acceleration to position, the constant vector added during integration only affects the components where acceleration is non-zero. For instance, if the acceleration is <0,1,0>, the initial x motion cannot be retrieved from the acceleration function alone. The integration constant, being a vector, does not provide additional information to determine the exact initial conditions. Without more data beyond the acceleration function, the specific initial state cannot be reconstructed. Understanding this limitation is crucial in multivariable calculus.
leehufford
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Hello,

In my multivariable calc class we are differentiating and integrating position, velocity and acceleration vector valued functions. My question is this:

When we integrate a vector valued function from acceleration to position, the constant vector only changes the definition of the functions that were not zero for acceleration. For example if the position function is <1,0,0> and the acceleration function is <0,1,0> , you could never recover that initial x motion from integrating the acceleration function.

What am I missing here? I understand the constant from integration is in the form of a vector now but this doesn't help me see what's wrong. Thanks for reading,

Lee
 
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This doesn't make sense because the position in the y component and z component are constant and 0, so there is no acceleration in this direction.
 
Any Two integrals are equal up to a constant term. The fact that the constant after integrating acceleration is a vector (velocity) doesn't mean that that integral is "more determined". You cannot recover exact situation unless given more than just acceleration function.
 

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