Transform Vector: Real Life Examples

[matqkks]

Why would we want to transform a vector in our normal basis (xyz axes) to another basis? The only situation I can recall is when we are looking at a force applied on an inclined plane. Are there any other real life examples where this may be necessary?

 

[arildno]

Why is what you call (xyz) our "normal" base?
For a box or any other object undergoing transformations, rotations and oscillations, perhaps the "normal" base is a set of axis fixed relative to its own geometry, rather than the base set up by an outside obersver?

And, for complex, "real-life" problems, you might encounter that som sub-aspect of the problem is most naturally expressed in one coordinate base, another sub-aspect most easily in another coordinate base.
Thus, the RATIONAL procedure is to develop a flexibility of the mind, and first formulate the sub-aspects in their most natural expressions, and THEN synthesize this into a common basis for calculations, utilizing general laws of coordinate transformation. therefore, you should learn about, and be adept at, those general laws as well.

By the way, if you aren't interested in deepening your understanding of the implications of the essential arbitrariness of a particular choice of base, and thus be motivated in how EVERY legitimate base choice are related to each other through laws of transformation, you really shouldn't bother study either maths or physics.

In short, you display the wrong attitude.