Velocity = length / time
so why a vector (x, y, z) is used to represent it?
+2
A:
Most likely to measure the change in three dimensional space for the object.
Magnitude of the vector should be the speed you expect, and as the object changes direction, the vector components will most likely change.
Michael
2009-05-19 00:20:42
+1
A:
You would use a vector because you can have velocity in 3 dimensions. In other words, the 3D velocity is the combination of distance/time in all 3 dimensions. It might be better to name the variables xPrime, yPrime, and zPrime, so that the vector more clearly represents velocity, rather than position.
Andy White
2009-05-19 00:21:17
Seriously, why did you vote this down? It's a perfect explanation. +1
Magnus Skog
2009-05-19 07:39:36
+10
A:
Because velocity is not "length/time". It is the first derivative of position. Position is a vector, and so its derivatives are also vectors.
kquinn
2009-05-19 00:21:52
Almost. Position is a point. Velocity is a vector. The difference between points are vectors, and so the derivative of a point is also a vector. People often don't make the distinction, and just treat vectors as equivalent to the point obtained by adding the origin to said vector.
Laurence Gonsalves
2009-05-19 00:44:58
Velocity is the first derivative of position with respect to time. It's a first order tensor in 2D or 3D space.
duffymo
2009-05-19 00:46:17
@kquinn: The essence of your answer is correct. I just wanted to point out that a point is not the same as a vector. Technically, you cannot add two points together (though affine combinations are okay), yet a point plus a vector yields a point and a vector plus a vector yields a vector. The distinction is subtle, but important and failing to understand the difference can lead to problems.
Naaff
2009-05-19 05:55:24
@kquinn: Position is not a vector! It is a point. You can represent a point by the nullvector pointing to it...
Kevin D.
2009-05-19 14:31:23
People saying "but position is a point, not a vector!" are missing the point. You need to define a coordinate system to do any kind of useful physics. Once you have a coordinate system, the position *vector* (usually just called the 'position', \vec{r}(t), is the vector from the coordinate system's origin to the point of interest. Bringing in distinctions between points and vectors is rarely useful, and bringing tensors into the whole mess is even less useful.
kquinn
2009-05-19 20:37:19
@kquinn: You may be right that you can often think of a point as a vector from the origin, but that doesn't mean that a point and a vector are the same thing. Promoting the false notion that a point and a vector are identical is not helpful. The two are distinct. And if you ever have to do anything non-trivial then problems *do* crop up. That's why there are whole spaces that unify points and vectors to avoid such problems in real-world situations (see projective space and Grassmann space). What point, exactly, are we missing? Wrong is wrong.
Naaff
2009-05-20 14:51:40
Did I ever say that a point and a vector are the same? No. I said that thinking of *position* as a point is not useful. When you're doing physics (and if velocity is involved, you're probably doing physics), you need a coordinate system. And when you have a coordinate system, the only kind of description of position that makes any sense is a position vector. Saying "but position is a point!" and "vectors are tensors!" and "affine spaces" *are not useful* to someone struggling to understand that the time derivative of a vector is also a vector.
kquinn
2009-05-20 20:49:04
An example: I have a point particle traveling through space. Maybe it's an electron, maybe it's the CM of a rigid body, whatever. What is its position? Saying "that point there!" is TOTALLY USELESS TO ANYONE. Saying "(0.3, 5.0, -1.3) in the lab coordinate system" provides a useful measurement of a position vector. If one second ago the particle was at (-0.7, 5.2, -1.7), then we can estimate its velocity vector as (1.0, -0.2, 0.5) -- again, a useful measurement. Points have positions. Positions are described by vectors. The time derivative of a vector is a vector. Don't overcomplicate it.
kquinn
2009-05-20 20:58:59
Nobody is overcomplicating anything. SO is a place for answers, and people deserve *correct* information. And saying "position is a vector" is wrong. Position is, and always will be, a point. *Change* in position is a vector. There is no such thing as a "position vector". In "(0.3, 5.0, -1.3) in the lab coordinate system" example (0.3, 5.0, -1.3) is a point. Points may *look* like vectors to you (they both have x,y,z components) but that doesn't make them vectors. I can translate a point. A vector is invariant to translation. I can add vectors. Adding points is undefined.
Naaff
2009-05-21 02:08:01
@kquinn: Nobody is trying to be hard on you or overcomplicate things. But in mathematics (and, by extension, physics) a "point" and a "vector" have very specific meanings. A vector has a magnitude and a direction (http://en.wikipedia.org/wiki/Euclidean_vector) while a point is a pure position with no length or direction (http://en.wikipedia.org/wiki/Point_(geometry)). While both can be represented with tuples like (x,y) or (x,y,z), they are honestly, truly distinct geometric objects. Please try to understand that all of the people who said a "point is not a vector" are only trying to help.
Naaff
2009-05-21 02:19:41
A:
Perhaps it is the speed that the object is moving in each of the directions in a 3D space, doing it this way means that you can extrapolate a direction of movement, after all velocity is movement with a direction.
Shraptnel
2009-05-19 00:23:49
+10
A:
Technically speaking, length divided by time gives you the speed, not velocity. Speed doesn't tell you which direction you are travelling in, while velocity does. In a three dimensional space, in order to describe where you are going and how fast, you need to supply three values: the direction AND speed you are going in each of the three fundamental directions (normally called axes and referred to by x, y, and z). But you could refer to them as forward/backward, sideways, and up/down if you want. For example, if you are travelling at 5km/hour upwards, the vector could be (0,0,5). Travelling 5km/hour downwards, your speed is just the same but the vector would be (0,0,-5). Travelling at 5km/hour at a 45 degree angle forward, the SPEED along each of the x and z axex would be the square root of 5, so the vector would be (approximately) (2.2,0,2.2). And so on.
dsteele
2009-05-19 00:32:01