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Points and Vectors

by Dan Sunday

 

 

Basic Definitions, Vector Addition, Scalar Multiplication, Affine Addition, Vector Length

Basic Definitions

A scalar represents magnitude, and is given by a real number: a,b...x,y....

A point in n-dimensional space is given by an n-tuple P=(p1,p2,...pn) where each coordinate pi is a scalar number. We will write P=(pi) as a shorthand for this n-tuple. The position of a point is relative to a coordinate system with an origin 0_n-coords and unit axes u1=10...0, u2=010...0, ..., and un=0...01. Thus, a 3-dimensional (3D) point can be given by a triple P=p1p2p3 whose coordinates are relative to the axes (1,0,0), (0,1,0), and (0,0,1) which are commonly referred to as the x, y, and z-axes. Because of this established convention, we sometimes write P=(x,y) for 2D points and P=(x,y,z) for 3D points.

A vector represents magnitude and direction in space, and is given by an n-tuple v=(v1...vn)=(vi) where each coordinate vi is a scalar. We also write v=(vi) as a shorthand for the vector's n-tuple. The vector v is interpreted to be the magnitude and direction of the line segment going from the origin 0_n-coords to the point Vpt=(v1...vn). However, the vector is not this point, which only gives a standard way of visualizing the vector. We can also visualize the vector as a directed line segment from any initial point P=(pi) to a final pointQ=(qi). Then, the vector from P to Q is given by:

Point Difference
Definition

v=Q-P=(qi-pi)

 

showing that the difference of any two points is considered to be a vector. So, vectors do not have a fixed position in space, but can be located at any initial base point P. For example, a traveling vehicle can be said to be going east (direction) at 50 mph (magnitude) no matter where it is located. We could even visualize a field of vectors, one at each point of a space; such as vectors for the wind direction and speed at each point on the surface of the earth.


Vector Addition

The sum of two vectors is given by adding their corresponding coordinates. So, for vectors v=(vi) and w=(wi), we have:

Vector Addition
Definition

vect_add_define


One can also add a vector v=(vi) to a point P=(pi) to get another point, given by:

Point+Vector Addition
Definition

Q=P+v=(pi+vi)


The resulting point Q is considered to be the displacement, or “translation”, of the point P in the direction of and by the magnitude of the vector v=Q-P.

Vector addition satisfies the following properties:

  • vect_add_assoc                [Vector Addition Association]
  • vect_add_commute                               [Commutation]
  • vect_add_point_assoc               [Point+Vector Association]

The property of “Point+Vector Association” means that result of translating a point by two sequentially applied vector displacements is the same as a single translation by the sum of those two vectors.

PIC_vect_P+(v+w)=(P+v)+w


Scalar Multiplication

Multiplication of a vector v=(vi) by a real number a (called a scalar) is given by:

Scalar Multiply
Definition

vect_scalar_mult_define

 

This represents scaling the size of a vector by a magnification factor of a. So, for example, 2v is twice the size of v, and v/2 = (1/2)v is half of v.

Scalar multiplication has the properties:

  • vect_scalar_assoc                 [Scalar Association]
  • vect_scalar_distrib           [Scalar Distribution]
  • vect_add_distrib         [Vector Distribution]

Scalar multiplication can be used to interpolate positions between two points P and Q. To get an intermediate point R between P and Q, given by a fractional ratio r, one first scales the vector v=Q-P to rv, and adds it to P to get: R=P+rv=... For example, to get the midpoint M between P and Q, use r=0.5 to compute M=0.5(P+Q).

This interpolation equation is also used to represent the line through P and Q as a function of a (1-dimensonal) scalar parameter, namely as: P=(1-t)P+tQ which is the parametric line equation. It is valid because one can use “affine addition” to combine points in a coordinate-free manner.


Affine Addition

We have already seen that the difference between two points can be considered as a vector. However, in general, it makes no sense to add two points together. Points denote an absolute position in space independent of any coordinate system describing them. Blindly adding individual coordinates together would give different answers for different coordinate reference frames.

Nevertheless, there is one special case, known as affine addition, where one can add points together as a weighted sum. In fact, in the previous section on scalar multiplication, we did just that to represent the points on a line going through two fixed points P and Q. More generally, given m points P0 , …, Pm–1, one can define:

Affine Sum
Definition

affine-sum-aiPi-define


where the coefficients, which can be any real number (positive or negative), must add up to 1. One interprets this sum as the center of mass for weights ai located at the points Pi. A negative coefficient can be thought of as a negative mass, like a helium balloon. This center of mass is uniquely determined in absolute space regardless of what coordinate system and frame of reference is being used. The affine sum has the property:

  • affine-sum-P+v=...                    [Translation Invariance]

Since the line equation is an affine sum, given equal weights a0=a1=0.5, we have that P(half)=half-P0+half-P1=half-(P0+P1) is the midpoint of the line segment from P0 to P1. Further, every point on the line through P0 and P1 is uniquely represented by a pair (a0,a1) with a0+a1=1, which results in the parametric line equation: P=(1-t)P0+tP1 where each point is represented by a unique real number t.

Similarly, in 3D space, the affine sum of three non-collinear points P0, P1, P2 defines a point in the plane going through these points. So, every point P in the plane of the triangle DELTA-P0P1P2 is uniquely represented by a triple (a0,a1,a2) with a0+a1+a2=1. This triple is called the barycentric coordinate of its associated point P on the plane with respect to DELTA-P0P1P2. Further, by substituting a1 = s, a2 = t, and a0=1-s-t, one can write the parametric plane equation as: P=(1-s-t)P0+sP1+tP2=..., where u=P1-P0 and v=P2-P0 are independent vectors spanning the plane.  The pair (s, t) is called the parametric coordinate of P relative to DELTA-P0P1P2, and there is a unique parametric coordinate for each point of the plane.


Vector Length

The length of a vector v is denoted by abs-v, and is defined as:

Vector Length
Definition

vect_len_define

 

This gives the standard Euclidean geometry (Pythagorean) length for the line segment representing a vector. For a 2D vector v=(v1,v2), one has: abs-v^2=v1^2+v2^2, which is the Pythagorean theorem for the diagonal of a rectangle.

Vector length has the following properties:

  • vect_len_assoc                               [Scalar magnification]
  • abs-(v+w).le.abs-v+abs-w                      [Triangle Inequality]
  • vect_Cosine-Law     [The Cosine Law, Euclid II, Prop 12&13]

A unit vector is one whose length = 1. One can scale any vector v to get a unit vector u that points in the same direction as v by computing: vect_unit=..., for which |u| = 1. The process of scaling v to a unit vector u is called normalization, and one says that v has been normalized. One thinks of u as the direction of v since v=abs-v.u simply scales u to the magnitude |v|.

 

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© Copyright 2012 Dan Sunday, 2001 softSurfer