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

In summary, using unit vectors allows for a clearer understanding of vector calculations by emphasizing the linear combination of basis vectors. It also helps to specify the coordinate system being used. While using ordered triples may be more compact, it can lead to a misconception that vector calculations are purely numerical rather than geometric.
  • #1
mujadeo
103
0
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??:confused:
 
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  • #2
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
 
  • #3
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 [itex](1, 2, 3) \cdot (4, 5, 6)[/itex] 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
 
  • #4
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.)
 
  • #5
Manchot said:
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.
 

1. How do unit vectors simplify vector calculations?

Unit vectors are vectors with a magnitude of 1 in a specific direction. By using unit vectors, we can break down a vector into its components in a specific direction, making calculations easier and more straightforward.

2. Why do we use unit vectors in vector calculations?

Unit vectors are used in vector calculations because they help us simplify complex vector equations. By using unit vectors, we can easily determine the magnitude and direction of a vector, as well as perform operations such as addition and subtraction.

3. What is the significance of unit vectors in physics?

In physics, unit vectors are crucial because they help us describe the direction and magnitude of a physical quantity, such as force or velocity. By using unit vectors, we can break down these quantities into their components, making it easier to analyze and solve physical problems.

4. How do unit vectors relate to the concept of basis vectors?

Unit vectors are a type of basis vector. Basis vectors are vectors that define a coordinate system and can be combined to represent any vector in that system. By using unit vectors as basis vectors, we can easily express any vector in terms of its components in a specific direction.

5. Can unit vectors be used in three-dimensional vector calculations?

Yes, unit vectors can be used in three-dimensional vector calculations. In three-dimensional space, we use three unit vectors (i, j, and k) to represent the x, y, and z directions, respectively. By using these unit vectors, we can break down any vector into its components in three-dimensional space and perform calculations with ease.

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