"We have focused our discussion on one-dimensional motion. It is natural to assume that for three-dimensional motion, force, like acceleration, behaves like a vector."- (Introduction to Mechanics) Kleppner and Kolenkow
We learn it very early in the course of our study that Force is vector; But, if I were the physicist defining the the Newton's second law (experimentally) and analysing the result F=ma, how would I determine whether Force is vector or scalar(especially in 3-D).
Actually, when I read the aforementioned sentences from the book, I wanted to know why do the authors expect it to be natural for us to think that in 3-D "Force" behaves like a vector. I know a(acceleration ) is vector and mass a scalar and scalar times vector gives a new vector but is there another explanation for this?
Answer
Uhm ... you start with an object at rest and notice that if you push on it in different directions it moves in different directions? Then notice that you can arrange more than two (three for planar geometries and four for full 3D geometries) non-colinear forces to cancel each other out (hopefully you did a force-table exercise in your class and have done this yourself).
The demonstration on an object already in motion is slightly less obvious but you can take the ideas here and generalize them.
In a sense this is so obvious that it's hard to answer because almost anything you do with forces makes use of their vector nature.
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