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Is there a method to "decompress" the acceleration scalar? For example, I am computing the scalar as: $\sqrt{a_x^2 + a_y^2 + a_z^2}$ where $a_x, a_y$ and $a_z$ are the components of the acceleration vector. If I have the scalar value (norm), is there a way to determine the $a_x, a_y, a_z$ components? A colleague mentioned using gyroscope or angular data, but did not know further.

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If I have the scalar value, is there a way to determine the x,y,z, components?

No. If all you have is this scalar, then you know the magnitude of the acceleration vector but you know nothing about its direction and you cannot determine its components. When you used the components to compute the scalar, you lost information. If you need the components, keep them.

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    $\begingroup$ There is one exception, when one can derive the components - when $|a| = 0$, then $a = \vec{0}$ :) $\endgroup$ Nov 11 '20 at 21:32
  • $\begingroup$ Good point. Thanks. $\endgroup$
    – G. Smith
    Nov 11 '20 at 21:33
  • $\begingroup$ @spiridon_the_sun_rotator That's assuming you are using a set of linearly independent components though. $\endgroup$ Nov 11 '20 at 21:38
  • $\begingroup$ @BioPhysicist and the norm of vector is a genuine norm, not a seminorm :) $\endgroup$ Nov 11 '20 at 21:39

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