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I think the word "projection" is, in general, used for both what you
call a component and what you call a projection.
For a couple of
examples where it is used to mean the number-with-units see:
<https://carnot.physics.buffalo.edu/archives/2002/09_2002/msg00392.html>
where it says:
" 'component' = as Bob says below, the projection of the vector of
interest onto the axes, what is traditionally labeled A_x and A_y
(signed scalar quantities)."
or see:
<http://www.av8n.com/physics/acceleration.htm>
where it says:
"The scalar acceleration can be considered one component of the vector
acceleration, namely the projection in the 'forward' direction (although
this is undefined if the object is at rest)."
Thomas Moore in his book /Six Ideas That Shaped Physics/ uses the
terminology "vector component" for the value-with-units and "component
vector" for what you call a projection.
Perhaps the use of the following terms would help avoid confusion:
vector component value
component vector
Note: I hesitate to use the word "scalar" for the vector component
value. I don't think of a vector component value as something that
transforms as a scalar (remains the same), for instance under rotation.
"What is the derivative of a scalar field, say dT/dx? Is it a scalar,
or a vector, or what? It is neither a scalar nor a vector, as you can
easily appreciate, because if we took a different x-axis, dT/dx would
certainly be different."