About writing vectors in physics.

[Timothy S]

I am confused as to how equations like F = mA would be solved. would you have to write the two Vector quantities in component form? And if not, how would a physicist turn the vector quantity in component form into a regular number for easy use?

 

[gsal]

Yeap, in this 3D world that we live in, we deal with vectors on a per component basis, as you say...one component for each physical dimension...i,j,k

There is no such thing as turning a vector into a regular number...a vector always has a direction and a magnitude...it is this magnitude that you are thinking as the "regular" number...and it is this number that you sometimes use but only in combination with other quantities that are in the same direction (i.e., vertical forces may be added up) or via the appropriate operation (cross-product, etc.,...refer to vector algebra).

So, yes, you represent vectors via their three orthogonal components, perform appropriate mathematical operations, then, at the end, when you want to know the final "regular" number you use the components to calculate the resultant of the quantity of interest.

 

[Timothy S]

wait, so in an equation can you write the final vector quantity as just a magnitude? for example Force would be X Newtons.

 

[PWiz]

When trying to describe a particular vector, you use the component vectors. When designating pure numbers to vector quantities, you can give the |modulus|. In general, when you're dealing with some 2D vectors, you can always split it into its component vectors and isolate the direction that matter using some trigonometry. When using equations involving vectors like $F = \frac{d}{dt} (\delta mv)$ , you must realize that the directions have not been taken into account (in the equation). So when you hear someone say "A force of 5N makes a mass of 2 Kg accelerate by 2.5 m s^-2 ", what they actually mean is an acceleration of 2.5 m/s^2 in the same direction as the force, but we don't really mention that since it is understood. Notice how I gave the vectors "force" and "acceleration" simple numbers with no mention of their respective components. So it all usually depends on the context of the question.

 

[Timothy S]

I see. So when want to calculate the force in that particular system you do not need the direction as it is not relevant information. But when direction is relevant, like when two different rocks of different momentum collide at a specific angle, you must use the components.

 

[PWiz]

Yes. When vectors are in the same directional orientation, you can simply carry out calculations by considering their magnitudes (similar to scalars).

 

[Nugatory]

wait, so in an equation can you write the final vector quantity as just a magnitude? for example Force would be X Newtons.

People do that a lot, but if you look carefully you'll see that there's also a direction involved and they just aren't mentioning it because it's (they hope) obvious from the context.

For example, if you're working a problem involving a cart moving down a straight road, you can choose your coordinate axes so that the x-axis is parallel to the road and the y and z components of all the force, acceleration, and velocity vectors are zero. In that situation we should say that the force vector is (for example) $\vec{F}=A\hat{x}+B\hat{y}+C\hat{z}$ with $B=C=0$... but it is soooo much more convenient to say that the force is just $A$.

 

[Timothy S]

I understand. But how would you manipulate the coordinate axes to make the collision along on axis?

 

[PWiz]

I understand. But how would you manipulate the coordinate axes to make the collision along on axis?

You can set the axes to your convenience. As Nugatory mentioned, if you're moving along a horizontal road you can define the x-axis as a line parallel to the road any particular coordinates above, below, or level with the line, simply because it's convenient. If I was observing the road from space or from an airplane window, I might "define" my axis in a different manner for ease of understanding. The problem comes when you have more than one observation in directions which don't agree with each other, which happens frequently : when a ladder rests against a wall, there is no "one axis" where all the directions agree. So we take at least one or two of the observations to agree with the axes, in this case by letting the surface be the x-axis and the perpendicular wall to be the y axis. Then you define the ladder position "relative" to these 2 axes.

 

[Timothy S]

Oh, I got it. So if you have a vector: F = 3x + 5y, how would you rearrange the axes in regard to P = 7x - 2y, so that you could describe the added vector as one number?

 

[Timothy S]

Also, what are you using to write the equations and symbols?

 

[Nugatory]

Also, what are you using to write the equations and symbols?

Latex. Try clicking the "Reply" link under my post and you'll see how I did it, and there's more documentation at https://www.physicsforums.com/help/latexhelp/

It's pretty easy, and even kinda fun once you get the hang of it.

 

[Nugatory]

Oh, I got it. So if you have a vector: F = 3x + 5y, how would you rearrange the axes in regard to P = 7x - 2y, so that you could describe the added vector as one number?

In that situation the two vectors are not parallel, so no amount of playing with the direction of the axes will allow them both to be described with a single number. You are stuck with using the machinery of vector arithmetic here. But it could be worse - you may not even have noticed, but both vectors do lie in the same plane, and you took advantage of that by using coordinates in which the the z component of both vectors is zero. So at least you're able to describe them with two numbers instead of the three you'd need if they weren't coplanar.

Collinear vectors: You can find coordinates in which you only need one number.
Coplanar vectors: You can find coordinates in which you only need two numbers.
General case of random vectors in three-dimensional space: You need three numbers per vector.