Excuse me jumping-in but here are a few thoughts.
@rudransh verma, try following these rules:
Put an arrow over the symbol for a vector, e.g. ##\vec V##. (But for unit vectors, use a ‘hat’, e.g. ##\hat i##.)
To represent the magnitude of a vector, use the unadorned letter, e.g. ##V##. Or if required for extra clarity, use ##|\vec V|##.
Never write equations where one side is a vector and the other side is a scalar. It is simply wrong.
I think the above is consistent with what Resnick does in your image in Post #45.
For example, take ##\hat j## as the positive upwards (+y) unit vector. Suppose there are two forces on an object, ##\vec A## up and ##\vec B## down. Then we can write these vector equations:
##\vec F_{net} = \vec A + \vec B##
##\vec F_{net} = A \hat j + B(-\hat j)##
##\vec F_{net} = (A - B)\hat j##
Since ##\vec F_{net}## has only a y-component we can write
##F_{net}\hat j= (A - B)\hat j##
Cancelling out ##\hat j## gives this scalar equation:
##F_{net} = A - B##
________________________
Also worth noting: the original problem in Post #1 should ask for the
minimum amount of work. This minimum would be achieved by raising the stone very slowly so that viscous (drag) forces and any increase in kinetic energy can be ignored.
Edit. Typo's corrected. Also, my post is a bit late as the thread has progressed.
