Intuition of 2D Velocity Vectors I understand 2D vectors in terms of displacement. For example, Bob moves 3m to the east and 4m to the north, the total change in position (displacement) equals 5m to the north east.
But I don't understand how 2D velocity vectors can be broken down into their horizontal and vertical components.
 A: Imagine a table, aligned to north-south-east-west just for convenience.  You have a velocity of 5m/s north-east (specifically at a bearing of 38.67 degrees, which is the correct angle for the 3-4-5 triangle you're choosing to use as an example).
Now look at it from the south edge of the table, so that all you can see is east and west  (technically also up-and-down, but we're not paying attention to up and down since everything is on the table).  If you looked at it from this perspective, you would see the object traveling at 3m/s east.  You couldn't see the north-south portion of its movement  because you're looking from the south edge of the table.
Likewise, if you viewed it from the east or west edge of the table, so that all you can see is the north and south movements, then you would see an object moving 4m/s north.
Now the reason we find this sort of splitting-up of vectors useful is that many of our equations for motion are linear functions.  This means that we can look at the north-south axis on their own, and look at the east-west axis on its own, and then combine the results to see the entire path the particle takes.  Your intuition may be having trouble with this because not all things you could want to look at have this nice linear function property to them.  If you're worried about something acting non-linear, then you're going to raise questions about why this splitting up was valid at all.  But rest assured, physicists have spent a great deal of time trying to make as many of their equations linear as possible, because it lets them do these sorts of clever vector tricks which make the math much easier!
A: Well, let's use the same case as you have used for displacement. If an object's horizontal vector component is 3 m/s to the right and the vertical velocity component is 4 m/s upwards, then the resultant velocity of the object must be 5 m/s to the north east corner of the graph.
Don't confuse the resultant velocity as the sum of the velocity components, because they don't all point in one direction. If you moved an object 3m to the right and 4m upwards, the total displacement from origin wouldn't be 7m, it would be 5m because the 3m and 4m aren't in the same direction. Think about the velocity vector in the same way. You can break it up into different components that point in different directions.

Now, you can break down the velocity into vector components in any arbitrary directions, but you usually break it up into horizontal and vertical because it helps you with solving problems. Particularly in problems where only a certain vector component matters or changes. Then, you don't have to worry about the velocity vector itself, only one (or more) of its components.
A: It works exactly like adding displacement vectors. The unit is not factored into the calculation of vector addition, only the magnitude, direction and sense. Your reasoning can be applied to velocity:
Bob moves at a rate of 3m/s to the east and 4m/s to the north, the total rate of change in displacement equals 5m/s to the north east.
Note(not exactly North East, 53° relative to the east, 37° relative to the north)
Also framing it as north, east, south and west can be limiting. The planes on which vectors are resolved do not have to represent anything .
A: We need to be very careful, in some cases, not to try to combine a horizontal and vertical vector into a single diagonal vector especially when they are incompatible like apples and oranges.  Take, for example, a tiny vector pointing and moving vertically.  When it is moving simultaneously horizontally it is still pointing vertically and just being transported horizontally.  At no time does it turn and point diagonally or horizontally.  That would be in violation of Newton's first law and would require a turning force to be applied.  The speed of light is fixed so if it is assigned to the vertical vector no other vector can be combined or added to it.  The horizontal vector does nothing to add to the forward momentum of the vertical vector.  The velocity of the vertical vector is fixed, the horizontal vector velocity is variable therefore they can not be combined into a time dilated diagonal vector.
A: Orthogonal axes make right angles, so you can use the Pythagorean theorem to get the hypotenuse. If you multiply the two shorter sides of a right angle by some unit transformation, or divide by a unit of time, the hypotenuse is identically scaled because of the Pythagorean theorem. It is Pythagoras's theorem that gives the meaning to orthogonal vector addition -- which holds regardless of units.
