Someone please explain why/how unit vectors help in vector calculations?

[mujadeo]

What is the benefit of using unit vectors rather than not using them??
I am not seeing the point of them? It seems to me that you can do the same calcs without adding the ihat and jhat.

Can someone explain exacly how and why unit vectors make vector calcs easier, or why we need them. Thanks

I do understand what a unit vector is, but i don't understand how they simplify calculations??

 

[cristo]

What do you mean by "simplify calculations"? Can you give an example of what you are talking about please? Remember that, when writing a vector as (a,b,c) you are using unit vectors, just not saying so, since this is equal to ai+bj+ck

 

[chroot]

The only advantage to explicitly writing the i, j, and k vectors is that it firmly impresses upon the student that all vectors are a linear combination of some set of basis vectors, and that remains true even when you manipulate them. That's the only reason.

Using notation like $(1, 2, 3) \cdot (4, 5, 6)$ is much more compact, but it can lull students into thinking that they're just playing games with numbers, when in fact they are doing geometry with vectors.

- Warren

 

[Manchot]

And, of course there's the other benefit that writing them explicitly tells you what basis vectors you're using. If you just write the ordered triple (1,0,pi), are you thinking in terms of Cartesian coordinates, cylindrical coordinates, spherical coordinates, or something else?

(My vector calculus book had a nasty habit of always writing things in terms of ordered triples, even when using non-Cartesian bases. It could be a pain.)

 

[cristo]

And, of course there's the other benefit that writing them explicitly tells you what basis vectors you're using. If you just write the ordered triple (1,0,pi), are you thinking in terms of Cartesian coordinates, cylindrical coordinates, spherical coordinates, or something else?

(My vector calculus book had a nasty habit of always writing things in terms of ordered triples, even when using non-Cartesian bases. It could be a pain.)

That must be pretty annoying, unless they state at the top of page what coordinates they're using or something.

I would always write (a,b,c) meaning cartesians and explicitly write in unit vectors if using any other "strange" coordinate system.