Does a component of a vector have components?

[titasdasplus]

TL;DR

Since component of a vector is a vector, the component should have component. Is it true? If yes,how?If no,why?

Since component of a vector is a vector, the component should have component. Is it true? If yes,how?If no,why?

 

[Ibix]

Welcome to PF.

I wouldn't say a vector component is a vector. Why do you think it is?

 

[kuruman]

TL;DR Summary: Since component of a vector is a vector, the component should have component. Is it true? If yes,how?If no,why?

Since component of a vector is a vector, the component should have component. Is it true? If yes,how?If no,why?

I would say that it is true. A vector can be considered to be the resultant (sum) of many vectors. These many vectors would be the "components" of the original vector in the sense that they add up (as vectors) to form it.

For 1D example, take a position vector that is 32 m long and points in the positive x-direction. You would write it as
$\mathbf L =32~(\text m)~\mathbf{\hat x}$.
Now cut it up into two equal pieces. You could consider the original vector, $\mathbf L$, as a vector which is the sum of two component vectors. You have
$\mathbf L =16~(\text m)~\mathbf{\hat x}+16~(\text m)~\mathbf{\hat x}.$
Cut each of the two pieces in half. Then you get a vector with four components
$\mathbf L =8~(\text m)~\mathbf{\hat x}+8~(\text m)~\mathbf{\hat x}+8~(\text m)~\mathbf{\hat x}+8~(\text m)~\mathbf{\hat x}.$
And so on.

The idea is no different from subdividing any number to a sum of numbers except that, here, there is a direction involved that must be the same for all the components. If a vector has components in orthogonal directions, e.g. $x$ and $y$, you would write
$\mathbf L =L_x~\mathbf{\hat x}+L_y~\mathbf{\hat y}$
and then subdivide $L_x$ and $L_y$ to as many pieces as you, in two separate directions instead of just one.

Of course, the same rule of simplification that applies to numbers also applies to vectors, you don't consider all the different ways that you can subdivide a vector into more component vectors in the same direction. You don't say "I have (7+3+11+8+3) dollars" instead of "I have 32 dollars." However, breaking a vector into two component vectors along two different directions is a useful tool when tackling physics problems.

 

[Dale]

Welcome to PF.

I wouldn't say a vector component is a vector. Why do you think it is?

I think that this can go either way. If I have a basis $\vec x$, $\vec y$, then I can write any arbitrary vector $\vec A$ as $\vec A = A_x \vec x + A_y \vec y$.

I have seen both $A_x$ and $A_x \vec x$ referred to as "the $x$ component of $\vec A$". The first case I think is more common among physicists and perhaps the second is more common among mathematicians (not sure about that). But the answer completely changes depending on which approach you take.

 

[titasdasplus]

Welcome to PF.

I wouldn't say a vector component is a vector. Why do you think it is?

Component means a part of a vector.So, I think Component is a vector .

 

[phinds]

Component means a part of a vector.So, I think Component is a vector .

Well, technically I think you are right, but it is sort of a vacuous truth. I mean, if you look at a vector going upwards to the right from the origin, it has an X component and a Y component, SO ... the X component is a vector going to the right horizontally and that vector has a zero Y component. True, but not particularly helpful.

 

[jtbell]

I have seen both Ax and Axx→ referred to as "the x component of A→".

My instinct is to call $A_x$ the “x component” and $A_x \vec x$ the “x component vector”.

 

[Dale]

If we mean $A_x$ as the component, then no, a component does not have components.

If we mean $A_x\vec x$ as the component (@jtbell 's "component vector", which I like), then yes, a component does have components.

The x component vector is itself a legitimate vector $\vec B = A_x \vec x$ and as such it can be legitimately written in component form as $\vec B = B_x \vec x + B_y \vec y$ where $B_x=A_x$ and $B_y=0$. It is also possible to write $\vec B = A_x \vec x$ in a different basis than the $\vec x$, $\vec y$ basis. This is valid mathematically, but physically it can easily lead to confusion.

 

[Andy Resnick]

TL;DR Summary: Since component of a vector is a vector, the component should have component. Is it true? If yes,how?If no,why?

Since component of a vector is a vector, the component should have component. Is it true? If yes,how?If no,why?

As some have pointed out, you can indeed expand (decompose) vectors into different basis vectors. A common example in Physics I is motion on an inclined plane, where the gravitational force vector g = g y is expanded into components perpendicular and parallel to the inclined plane, say g cos(θ) u + g sin (θ) v.

 

[Ibix]

Component means a part of a vector.So, I think Component is a vector .

If you have a vector $\vec{A}=A_x\vec{x}+A_y\vec{y}$, as @Dale's notation, then I would call $A_x$ and $A_y$ "components", and these are not vectors. However, others are also correct that "component" is sometimes used for $A_x\vec{x}$ or, more generally, $(\vec{A}\cdot\hat{\vec{v}})\hat{\vec{v}}$. That is, indeed, a vector.

So the question is, what do you (or your professor, or your textbook) mean by "component"? Just the number, or the number multiplied by a unit vector? In the first case, a component is not a vector. In the second case, yes it is.

 

[ok123jump]

All of the above answers are great. The answer is in almost every definition, yes. A vector can be written as a sum of unit vectors, and those are still vectors. They are just 0 in all but a single direction.

 

[phinds]

They are just 0 in all but a single direction.

ALL vectors, taken only in themselves, are just 0 in all but a single direction. Things depend on your coordinate system, so to be fully accurate, one has to say that "... those are still vectors. They are just 0 in all but a single direction, IN THE COORDINATE SYSTEM OF THE ORIGINAL VECTOR.

 

[ok123jump]

ALL vectors, taken only in themselves, are just 0 in all but a single direction. Things depend on your coordinate system, so to be fully accurate, one has to say that "... those are still vectors. They are just 0 in all but a single direction, IN THE COORDINATE SYSTEM OF THE ORIGINAL VECTOR.

That’s true, but given the nature of the question, I think it’s a bit too much to add this level of detail right now. I don’t think he’s working in different coordinate systems if he doesn’t have a solid understanding of unit vectors - let alone basis vectors.

You aren’t wrong. I just question whether OP’s ready for this level of detail.

Side Note: This is the problem with much of Wikipedia, btw. I wrote a couple articles to be easily understood by undergrads, and some pretty sophisticated postdocs came through with specific definitions and made them more precise, but wholly inaccessible to anyone but other postdocs. That happens all across that platform. Just trying to make sure we don’t get dragged into that quagmire here.

 

[phinds]

Yes, good point.

 

[titasdasplus]

My instinct is to call $A_x$ the “x component” and $A_x \vec x$ the “x component vector”.

I support your idea.