If a 3D vector represents a point, how can it have a length?

Question

I am trying to understand vector arithmetic (and specifically its use in the Unity engine). I am not able to figure out how a vector can have a length (magnitude) even though it only represents a point (position and direction)?

Does that mean that the magnitude is simply its distance from the origin point (0, 0, 0)? Or am I missing something?

Answer 1

Does that mean that the magnitude is simply it is distance from the origin point (0, 0, 0)?

The tl;dr answer may be: Yes, you can imagine it like that.

But I'm not sure whether this might not lead to a wrong understanding.

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A vector is not a point, and there is a crucial difference between the two!

The fact that a vector is usually represented as an "arrow" might give a wrong impression. A vector is, in fact, not a single arrow. It would be more precise to say that a vector is the set of all arrows that have the same length and direction. (The arrow that is usually painted is just one representative of all these arrows). But I don't want to go too far into the boring details of mathematics here.

More importantly, there is a crucial difference between a point and a vector, that becomes obvious in graphics programming when you transform the point or vector. I'm not familiar with Unity, but from a quick glance at the documentation, they are modeling the most important difference between a point and a vector in the Matrix4x4 class. It has two different functions:

• Matrix4x4.MultiplyVector and
• Matrix4x4.MultiplyPoint

The difference is, roughly speaking, that a vector is not translated, whereas a point is. Imagine the following 4x4 matrix:

1.0   0.0   0.0   1.0
0.0   1.0   0.0   2.0
0.0   0.0   1.0   3.0
0.0   0.0   0.0   1.0

It describes a translation about (1,2,3). Now, when you have the following pseudocode

Vector3 tp = matrix.MultiplyPoint (new Vector3(2,3,4));
Vector3 tv = matrix.MultiplyVector(new Vector3(2,3,4));

Then tp will be (3,4,5), wheras tv will still be (2,3,4). Translating a vector does not change it (because, as mentioned above, it is the set of all arrows with the same magnitude and direction).

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The fact that Unity uses the Vector3 class for both, vectors and for points, is legitimate, but may be confusing. Other libraries dedicatedly differentiate between Point3D and Vector3D, sometimes with a common base like Tuple3D.

Answer 2

Does that mean that the magnitude is simply it is distance from the origin point (0, 0, 0)?

That's exactly that.

Among other things, a vector can represent a point (a position), a direction and/or a velocity, depending on the context.

If you have this variable:

Vector3 mPosition;

It generally represents only the position, i.e. where it is located in 3d space.

If you have this variable:

Vector3 mDirection;

It generally represents the direction. Typically, these vector are unit vectors, i.e. vectors of length 1 (but it's not always needed). A unit vector and a Normalized vector are the same thing, they're both of length 1. These vectors are often used with other vectors to change their positions.

When normalizing a vector, you lose its length (its magnitude), but the direction stays the same. There are situations when you only need the direction (e.g. when you want to move an object in that direction), and having the (non-unit-length) magnitude in the vector would introduce unexpected calculation results.

If you need a normal vector for a single calculation, you can use myVec3.normalized, it will not affect myVec3, and if you intend to use that normalized vector often, you should probably create a variable:

Vector3 myVec3Normalized = myVec3.normalized;

to avoid repeated calls to the normalized method.

And if you see variables:

Vector3 mVelocity;

It generally represents a force/speed: these vectors represent a direction and their magnitude (their length) is important. They could also be represented with Vector3 mDirection; and a float mSpeed;.

All of these are used with regard to their local origin, which can be (0, 0, 0), or can be another Position.

Answer 3

Does that mean that the magnitude is simply it is distance from the origin point (0, 0, 0)?

You can see it that way, but only seeing it that way may lead to a wrong understanding.

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First of all, a vector is not a point, and a point is not a vector.

The difference between a vector and a point is the same as between a duration and a time of day. The former is an interval of time, the latter is a single point in time. Its obviously that 6 hours is not the same as 6 o'clock. You wouldn't say "The race lasts 1 o'clock" and neither would you say "Lets meet at 13 hours". The race lasts one hour - an interval - and you meet at 13 o'clock - a specific point in time.

The same applies to vectors and point. A vector is an intervall - a displacement if you will. It points in a certain direction, and yes, it has a length.

Points and vectors are therefore related, just as durations and times of day. The race starts at 13 o'clock and ends at 15 o'clock. Both are points in time. But 15 o'clock - 13 o'clock = 2 hours, a duration. The race lasts two hours, not 2 o'clock.

The same applies to points. The difference between point A and B is denoted as ⃗v = B - A, where ⃗v denotes a vector and A and B denotes points.

Now, there is something that called a position vector. You can consider a vector a point to a certain degree, when you say that the vectors points from the origin to a certain other point. In other words: If all your friends know that you call times of day as durations since midnight (0 o'clock), you can say "We meet at 6 hours". They would know that 0 o'clock + 6 hours = 6 o'clock and therefore, when to meet you. This is in fact what naval times do. "We meet at o-six-hundred hours" means 6 o'clock.

So the vector <1,2,3> points to the point (1,2,3), if you consider the origin the anchor point, and yes, the length of this vector is the distance of that point from the origin.

But the vector <1,2,3> also points from (1,1,1) to (2,3,4), and in that case its length denotes the distance between those two points.

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So, as you can see a vector has a length because it is not a point, but an interval -- a displacement.

Answer 4

A vector can represent a line between two points in 3d space (direction and distance) or a location in 3d space (length is the distance from origin).

If you have point A, and point B, then B-A = AB = the direction and distance you would have to travel to get from A to B.

Answer 5

What Unity says about points vs. vectors is meaningless in the long run, because geometry APIs just pick distinct definitions to make the tool more accessible, they don't correspond to how these things are conceptualized in geometry. Take a look at the implementations of the classes, if you can. Because it's arbitrary, to know its definition is the only way to understand what the concept is. Full disclosure, I don't have Unity experience.

A vector is a point in a vector space, in that the concept of a point in the geometry is encoded by elements of the underlying set. A vector space has a distinguished vector, called the origin or 0. Linear algebra is an attempt to encode a fragment of euclidean geometry w/ an origin algebraically.

The arrow and its length

Motions across a space of points are frequently interpreted as all the arrows from source/before points to their target/after points.

A function of two arguments can be applied to one argument to produce a function of one argument - we can speak of x+, the function which takes each vector y to the vector x+y. This is the translation associated w/ adding x. The associated arrows run from points y to points x+y. See: partial application, currying.

So why do we only use the one arrow? The arrow from the origin points to a specific vector, the x in x+ - the origin is the identity of vector addition. So, we can recover the translation x+ from just its value x+0 = x.

As a graphical representation of the space, the arrow representation has to do with our ability to visually or physically extrapolate the effect of a translation from the value determining it. When do we have that ability?

To give the vector space a norm making it a normed vector space is to provide a notion of the length of a vector that makes sense as its distance from 0. As well, this is to be a distance satisfying the triangle inequality, which is a strong constraint on how the lengths of two vectors relate to that of their sum. From length we can define distance to make this a metric space, and a geodesic is a path that's intrinsically straight in that it's as short as possible. The euclidean norm induces euclidean distance and the geodesics are the line segments of the arrows, but if you draw the arrows as geodesics using different norms, you could extrapolate the geometric effect of the translation from the geodesics to learn about the geometry.

The meaning of point and vector

In some cases in doing games geometry, your space of points is not a vector space. An affine space of dimension n can be embedded in a projective space of dimension n. Affine maps reduce to projectivities. Projectivities also let you do FOV, w/c I think is not affine. Projectivities have benefits:

The projective n-space over a field can be constructed from the linear (n+1)-space (vector space), by treating the points of the projective space as the lines through the origin of the linear space. Planes through the origin in turn give projective lines. Multiplying vectors by a fixed matrix is a linear map, this is what matrix multiplication is for. Linear maps preserve the origin and are compatible w/ incidence. In particular, if f is a linear automorphism (corresponding to an invertible (n+1)x(n+1) matrix), and two lines L, M through the origin span a plane A, then f L, f M are lines through the origin spanning f A, so f will preserve incidence on the projective space as well - an invertible matrix has an associated projectivity. Matrix multiplication encodes composition of linear maps, and hence of the projectivities.

Removing the origin from the linear space, all the points on a given line through the origin are scalar multiples of one another. Exploiting this fact, homogenization picks a linear point to stand in for each projective point and an invertible matrix to stand in for each projective transformation (as in this 2D -> 2D affine maps as 3D -> 3D linear maps video), in such a way that the representatives are closed under matrix-matrix and matrix-vector products and give and are given by unique projective things. This description of construction of the projective plane from the linear plane ties some things together.

So, in the model-view-projection matrix pipeline, we're using vectors to represent the points of our projective space, but the projective space is not a vector space, and not all vectors in the vector space we're using represent points of our geometry (see picture of affine plane at right). We use translation matrices instead of vector sum if we want translations. Sometimes, people call projective or affine points vectors, especially when using a setup in this vein.

Answer 6

The length (or magnitude) of the vector is square root of (x*x+y*y+z*z). Vectors are always considered as a ray passing from the origin of <0,0,0> through the point described in the vector <x,y,z>

The unity documentation on this is found here.