Sign problems with vectors, how can we "resolve" this....

[etotheipi]

This business of positive and negative values is not the issue. In terms of the physics, you start with the concept that a vector has, in some physical sense, a direction. In classical physics, that means a direction in 3D space. Mathematically this is called a directed line segment. In that sense, a vector is defined by a directed line segment.

Right, though in that case even for one dimensional motion we'd require $\vec{F} = m\vec{a}$, where $\vec{F} = F\hat{x}$ and $\vec{a} = a\hat{x}$. First we need to get all of our forces in terms of $\hat{x}$, which essentially defines the positive direction along the chosen axis.

I suppose then you could "cancel" $\hat{x}$ and end up with an equation containing $F$ and $a$. And by playing around for a little bit (proof by a few examples... my favourite method), I think it's the case that anyone dimensional situation can be described completely in terms of the components. Same goes for rotational dynamics; I don't often write $\theta \hat{z} = \omega t \hat{z}$, but just use the component terms.

In this vein, I don't think it makes sense to simply state $F_{x} = ma_{x}$ on its own without 'obtaining it' from the above sort of logic. Otherwise, the two quantities are completely undefined - the components cannot exist without the force vector!

 

[Herman Trivilino]

Once we introduce the vector object, these scalars just become the components and contribute to the overall machinery.

No, they were components all along. It's just that it's usually not made explicitly clear in the interest of brevity.

For example, the equation $v=v_o+at$ that is used in the study of one-dimensional motion should be written, for example, as $v_x=v_{ox}+a_xt$ to make it explicitly clear that we are talking about vector components and not vector magnitudes. Of course, that introduces an extra layer of complexity for students who are already struggling to understand things.

 

[etotheipi]

No, they were components all along. It's just that it's usually not made explicitly clear in the interest of brevity.

For example, the equation $v=v_o+at$ that is used in the study of one-dimensional motion should be written, for example, as $v_x=v_{ox}+a_xt$ to make it explicitly clear that we are talking about vector components and not vector magnitudes. Of course, that introduces an extra layer of complexity for students who are already struggling to understand things.

Yeah, I now realize that I was completely bluffing it in that post, now that I re-read it it's a little embarrassing.

Your post makes it clear

 

[robphy]

I think a big part of the problem is notation... and possible confusions when generalizing to high dimensions.

Right, though in that case even for one dimensional motion we'd require $\vec{F} = m\vec{a}$, where $\vec{F} = F\hat{x}$ and $\vec{a} = a\hat{x}$.

The more correct expressions [and less ambiguous expressions] are
$\vec{F} = F_x\hat{x}$ and $\vec{a} = a_x\hat{x}$... (where $F_x \equiv \hat x\cdot \vec F$)
since $F$ (in the context of vectors) is the magnitude of $\vec F$.
If you want to use $F$, then you can write $\vec F = F \hat F$ (where $F\equiv\sqrt{ \vec F\cdot\vec F}$ and $\hat F\equiv \vec F/F$).
The physical law is \vec F=m\vec a.

In a coordinate system, its components are F_x=m a_x, F_y=m a_y, etc... (where F_x=\vec F\cdot \hat x, etc... ).
So, \vec F=ma_x \hat x + m a_y \hat y + \cdots
In one-dimensional problems, this becomes:
\vec F=ma_x \hat x, or in component form F_x=ma_x.

Usually, since books start out in 1-D, then generalize to higher dimensions,
this component equation F_x=ma_x is
often written as F=ma (for simplicity, since F_x=ma_x is too complicated?),
where F and a are really signed $x$-components of force and acceleration,
although the phrase "signed $x$-components of" is usually omitted.

Then, generalizing to higher-dimensions,...
in many physics books, the convention is "a vector \vec F with its arrowhead removed" or "a vector {\bf F} unbolded",
F represents the vector magnitude:
F= | \vec F |= \sqrt{\vec F\cdot\vec F}
or F= | {\bf F} |= \sqrt{{\bf F}\cdot{\bf F}} .
So, \vec F = F \hat F...(for simplicity, because \vec F = |\vec F| \hat F looks too complicated?)
but this "F" meaning "|\vec F|" (the non-negative magnitude) here
is different from "F" meaning "F_x" (the signed component along \hat x) in the previous paragraph.

So, there is now a clash in notation... since $F$ has been doing double-duty to avoid complication(?).

 

[etotheipi]

I think a big part of the problem is notation... and possible confusions when generalizing to high dimensions

The more correct expressions are ...

Wow, this is a great reference resource - retroactively helps me to understand a lot of what has already been said in the thread that I hadn’t totally gotten yet! Thanks!

 

[etotheipi]

For example, the equation $v=v_o+at$ that is used in the study of one-dimensional motion should be written, for example, as $v_x=v_{ox}+a_xt$ to make it explicitly clear that we are talking about vector components and not vector magnitudes. Of course, that introduces an extra layer of complexity for students who are already struggling to understand things.

Just one further thought; it definitely seems clearer to insert the subscripts in order to distinguish components like $v_{x}$ from magnitudes like $v$, however when I look at even some undergraduate level lecture notes it seems the latter is used quite often also for the signed component - the notation clash @robphy referred to.

I wonder whether in your opinion it would be worth getting into the habit of writing out the subscripts (which is what I'm sort of inclined to start doing, purely for clarity's sake) even for one dimensional motion?

The downside is evidently brevity, and if the system perhaps requires more than one coordinate system when solving (e.g. pulleys) we might need to start putting in $x'$'s and $y'$'s et cetera which takes a little longer. Though it arguably reduces any ambiguity.

What do you think?

 

[Herman Trivilino]

I wonder whether in your opinion it would be worth getting into the habit of writing out the subscripts (which is what I'm sort of inclined to start doing, purely for clarity's sake) even for one dimensional motion?

The downside is evidently brevity, and if the system perhaps requires more than one coordinate system when solving (e.g. pulleys) we might need to start putting in x′x′x''s and y′y′y''s et cetera which takes a little longer. Though it arguably reduces any ambiguity.

What do you think?

You can look at what Randy Knight does in his introductory textbooks. He uses a subscript of $s$ to stand for any component in general, and then replaces $s$ with $x$ for horizontal motion and with $y$ for vertical motion. Sounds neat, but we also have to deal with inclined planes.

Since you understand the role played by the subscripts it's probably a good idea for you to insert them. I have tried teaching it both ways, and prefer to leave the subscripts off at the beginning. They can be inserted later on in the introductory course when the study of two-dimensional motion begins.