When can you add vectors? I was wondering when it was okay to just add two vectors together. Like, adding two velocity vectors will give you the overall velocity vector. Does this apply to acceleration as well? Can you find acceleration in the $x$ and $y$ direction, and then find the overall acceleration at that point?
I remember encountering a time when I couldn't just add two vector quantities together to get the overall vector.
Could someone please clarify this?
 A: You can add whatever vectors you want as long as you put them in the same coordinate space first. That is to say, before we can add x-components to x-components, we have to agree that x is a direction and which direction it is.
In physics, we care about describing real things, not unitless values, so you have to make sure the vector sum describes what we want to describe.
Even so there's nothing except stopping you from adding irrelevant things together except that you'd be wasting your time. They don't even need to have the same units - just make sure you carry your units all the way through: 2+2 = 4, but 2L + 2m doesn't equal 4 of anything.
You can add the gravitational attraction applied on you by the moon when it's directly overhead to the velocity of your cat with respect to the mailman. The answer will be something like $2\times 10^{-3} N \hat z + 5m/s \hat x + 2 m/s \hat y - 1 m/s \hat z$. What does it mean? Absolutely nothing.
A: Well, yes. Recall that during circular motion, we used to find net acceleration using vector sum of accelerations.
So it js possible to add two vectors of same quality.
Shall you please share the point where you are unable to add two vectors?
A: This topic can get very complicated because vectors are so heavily used, hence there is so many situations that can create problems.
The two check I would do are:
(1) Same Units
Vectors certainly need to be of the same units before it makes sense to add them
(2) Same Place/Time
Vectors normally should be 'in the same place' before you should add them.
This second bit is more subtle. For example lets say you have electric field $\mathbf{E}_a$ at point $a$ and electric field $\mathbf{E}_b$ at point $b$. Should you be adding them? I would say no. If there is a physical scenario which calls for it, check the scenario - something is wrong. It is however possible that you have a procedure for converting both electric fields into scalars and then adding them, e.g. with a surface integral $\int_\Omega d^2s \mathbf{\hat{n}}.\mathbf{E}$, where $a,b\in\Omega$.
Why is it a problem to add vectors in different places? Well, it is because definition of a vector in space relies on the tangent space of the manifold you are in, and different places on that manifold have different vector spaces. In normal 3d space this often makes no difference. But consider adding a vector pointing 'up' for you in, say, Europe, and a vector pointing 'up' for someone in Australia. Does it make sense? Not really, curvature of Earth will mess it up. So if you encounter a physical situation that calls for such, potentially problematic, procedure, be sceptical
A: We can have the same doubt about scalar quantities. A steel shape like an angle has 2 legs. Its mass is the sum of the mass of each leg, but its temperature is not the sum of the temperatures of each leg. Not all quantities result from adding smaller parts.
The net force on the angle is the sum of the forces on each leg. We add vectors in this case instead of scalars. But as we can not add temperatures, it is also not possible to add the velocities of each leg to get the overall velocity of the shape.
