Difference between a vector in maths and programming

Question

Maybe this question is better suited in the math section of the site but I guess stackoverflow is suited too. In mathematics, a vector has a position and a direction, but in programming, a vector is usually defined as:

Vector v (3, 1, 5);

Where is the direction and magnitude? For me, this is a point, not a vector... So what gives? Probably I am not getting something so if anybody can explain this to me it would be very appreciated.

Answer 1

If we are working in cartesian coordinates, and assume (0,0,0) to be the origin, then a point p=(3,1,5) can be written as

where i, j and k are the unit vectors in the x, y and z directions. For convenience sake, the unit vectors are dropped from programming constructs.

The magnitude of the vector is

and its direction cosines are

respectively, both of which can be done programmatically. You can also take dot products and cross-products, which I'm sure you know about. So the usage is consistent between programming and mathematics. The difference in notations is mostly because of convenience.

However as Tomas pointed out, in programming, it is also common to define a vector of strings or objects, which really have no mathematical meaning. You can consider such vectors to be a one dimensional array or a list of items that can be accessed or manipulated easily by indexing.

Answer 2

In mathematics, it is easy to represent a vector by a point - just say that the "base" of the vector is implied to be the origin. Thus, a mathematical point for all practical purposes is also a representation of a mathematical vector, and the vector in your example has the magnitude sqrt(3^2 + 1^2 + 5^2) = 6 and the direction (1/2, 1/6, 5/6) (a normalized vector from the origin).

However, a vector in programming usually has no geometrical use, which means you really aren't interested in things like magnitude or direction. A vector in programming is rather just an ordered list of items. Important here is that the items need not be numbers - it can be anything handled by the language in question! Thus, ("Hello", "little", "world") is also a vector in programming, although it (obviously) has no vector interpretation in the mathematical sense.