Question

please i dont really understand the concept of angular displacement not being a vector,need some detailed begginers explanation please

Answer 1

When you're first introduced to scalars and vectors, you're usually told that the difference between them is that vectors have a specific direction but scalars do not. This is usually an adequate distinction if you're just using them at an introductory level.

Although this definition of a vector is often true, it's not solely what makes a vector a vector. Vectors must obey several different principles, one of which is known as "commutation". This is a fancy word meaning if you have two values, the ORDER in which you add/subtract/multiply/divide/etc these values does not affect the result; the values are commutative.

As an example in the context of vectors, you can go 10 km west and then 5 km north, or 5 km north and then 10 km west. Either way, you end up at the same spot. A+B=B+A. The order you added these linear displacement vectors did not change the value of the sum. This is part of what makes linear displacement a vector quantity.

Angular displacements, on the other hand, are not considered vectors because they are not commutative. Say you have a book on the table in front of you lying face up, and you want to rotate it twice:

A. 90 degrees clockwise, about an axis perpendicular to the table
B. 90 degrees clockwise, about an axis perpendicular to you

After the first rotation, the book is still face up in front of you, but sideways, such that its spine is facing away. After the second rotation, the book is now standing upside down on its edge, with the cover facing off to your right and the spine still pointing away from you. Now let's reset the book, and perform the rotations in the reverse order:

B. 90 degrees clockwise, about an axis perpendicular to you
A. 90 degrees clockwise, about an axis perpendicular to the table

After the first rotation, the spine of the book is pointing straight up in the air, with the cover facing your right. After the second rotation, the spine is still pointing up in the air, but the cover is now facing you!

You can see that the book ends up in a DIFFERENT position depending on the order you add the angular displacement. A+B does not give the same result as B+A. Angular displacement is NOT commutative, so it is not truly a vector! Try doing the steps above with a book yourself so you can see what I mean!

Hopefully that clarifies things, but if you still have questions let me know!

Answer 2

excellent, thanks matt