Vector maths basics

[JH12]

I'm learning a bit about basic vector maths. 
I'm getting a bit confused by vector direction. My understanding thus far is that a vector will always get it's direction based on its relationship to origin (i.e. draw a line between 0,0 and x,y and that's your vector direction).  But then I have seen some Houdini tutorials which talk about adding one vector to the tip of the other to get a vector which has a direction which is not based on its relationship to origin, but rather to the tip of the previous point.  Is this possible, and if so how is this determined, given that a vector only has the coordinates of a single point in space?
Example in my attached picture of A+B, where I was told that the result would be the green vector. Though based on my understanding of vectors I would have thought this would always result in the dotted blue line.

    

[toadstorm]

The dotted blue line is mathematically correct. The result of adding those two vectors results in a third vector, but it's still going to be relative to the parent space of those two vectors, meaning relative to the origin.

[Yon]

yea, whoever said that is incorrect. the green line labeled A+B here is just B placed on the tip of A. and the dotted blue line is A+B


vector basics

 

[kiryha]

Vector is just a direction and length and it does not matter where the vector is located in Cartesian space (there is a special type of vector called "position vectors" which lock to space and always starts at the origin, though).
So, vector B is the same vector as green A+B. Placing B on the tip of A helps to visualize that A + B = blue line.

[JH12]

5 minutes ago, kiryha said:

Vector is just a direction and length and it does not matter where the vector is located in Cartesian space (there is a special type of vector called "position vectors" which lock to space and always starts at the origin, though).
So, vector B is the same vector as green A+B. Placing B on the tip of A helps to visualize that A + B = blue line.

But how would you define the direction of the green vector, because without the point of A as a reference, how are we to know by reading the green vectors coordinates that the origin is the tip of A?

[kiryha]

18 minutes ago, Fireandsmoke said:

But how would you define the direction of the green vector

To define a vector you need to define how far it travels in X and Y direction. For vector B, roughly from your picture, it would be 1 in X and 5 in Y (B = [1, 5]). Vector A would be 3 in X and 2 in Y (A = [3, 2]). And it's just two vectors which don't have any relation between them (its just two pairs of numbers).
If you need to add this vectors you will add X and Y values, so A + B = [1+3, 5+2] = [4, 7] (blue line)
Moving vector A to a position of a green vector (called A+B which is wrong, cos its the same vector A, not A+B) just helps to illustrate the math we done: the blue line is a result of adding  A and B

[CinnamonMetal]

Vectors always start at the origin, that is how you get the length when adding two vectors.

[kiryha]

Relying on the math for VFX course, only position vectors start at the origin. So I would not say always

[StepbyStepVFX]

You can also look at :

https://www.fxphd.com/details/215/

https://www.fxphd.com/details/389/

 

[Jesper Rahlff]

this playlist of videos have a lot of good and well explained theory about vectors:

 

 

[malexander]

You can also think of vectors in 1D - just a simple number line. 2+3 = 5. You start at zero and go to 2 (A). B then goes 3 units to the right (2->5). So the total vector is 5. You original example just extends that number line into 2 dimensions (2 number lines. which you can solve independently), and if you're dealing with 3D vectors, then you get an additional number line.