Kuzee
Vector Algebra and Analytic Geometry of Space

Properties and Applications of Vector Operations

The Basic Properties of Vector Operations

Properties of Addition

The basic properties of vector addition

The commutative law: a⃗+b⃗=b⃗+a⃗\vec{a} + \vec{b} = \vec{b} + \vec{a}

The associative law: (a⃗+b⃗)+c⃗=a⃗+(b⃗+c⃗)(\vec{a} + \vec{b}) + \vec{c} = \vec{a} + (\vec{b} + \vec{c})

The zero vector: a⃗+0⃗=a⃗\vec{a} + \vec{0} = \vec{a}

The opposite vector: a⃗+(−a⃗)=0⃗\vec{a} + (-\vec{a}) = \vec{0}

Properties of Scalar Multiplication

The basic properties of scalar multiplication

Distributivity: k(a⃗+b⃗)=ka⃗+kb⃗k(\vec{a} + \vec{b}) = k\vec{a} + k\vec{b}

Distributivity over scalar addition: (k+l)a⃗=ka⃗+la⃗(k + l)\vec{a} = k\vec{a} + l\vec{a}

Associativity of multiplication: k(la⃗)=(kl)a⃗k(l\vec{a}) = (kl)\vec{a}

The identity element: 1⋅a⃗=a⃗1 \cdot \vec{a} = \vec{a}

Properties of Mixed Operations

The mixed properties of vector operations

Distributivity: k(a⃗+b⃗)=ka⃗+kb⃗k(\vec{a} + \vec{b}) = k\vec{a} + k\vec{b}

Distributivity over scalar addition: (k+l)a⃗=ka⃗+la⃗(k + l)\vec{a} = k\vec{a} + l\vec{a}

Associativity of multiplication: k(la⃗)=(kl)a⃗k(l\vec{a}) = (kl)\vec{a}

Mixed operations: (k+l)a⃗+ma⃗=(k+l+m)a⃗(k + l)\vec{a} + m\vec{a} = (k + l + m)\vec{a}

The Basic Theorems of a Vector Space

The closure of vector operations

A vector space is closed under vector addition and scalar multiplication, that is:

  • the sum of any two vectors is still a vector
  • the product of any vector with a scalar is still a vector
The uniqueness of vector operations

In a vector space, the results of the addition and the scalar multiplication are unique.

Practical Applications

Applications in Physics

Mechanics Analysis

In mechanics, the computation of the resultant force when several forces act at the same time:

F⃗total=F⃗1+F⃗2+F⃗3+⋯\vec{F}_{total} = \vec{F}_1 + \vec{F}_2 + \vec{F}_3 + \cdots

The Composition of Displacements

The total displacement of an object is the vector sum of the displacements of each stage:

s⃗total=s⃗1+s⃗2+s⃗3+⋯\vec{s}_{total} = \vec{s}_1 + \vec{s}_2 + \vec{s}_3 + \cdots

Applications in Computer Graphics

3D Transformations

In computer graphics, vector operations are used to implement the transformations of 3D objects:

  • translation: vector addition
  • scaling: scalar multiplication
  • rotation: involves more involved vector operations

Ray Tracing

The computation of a ray’s direction involves vector operations:

r⃗=o⃗+td⃗\vec{r} = \vec{o} + t\vec{d}

where o⃗\vec{o} is the position of the light source, d⃗\vec{d} is the direction of the ray, and tt is the distance parameter.

Applications in Engineering

Structural Mechanics

In the design of bridges and buildings, vector operations are used to analyse the forces on a structure:

  • computing the support reactions
  • analysing the stress distribution
  • determining the stability of the structure

Fluid Mechanics

In fluid mechanics, the analysis of the velocity field:

v⃗=v⃗xi⃗+v⃗yj⃗+v⃗zk⃗\vec{v} = \vec{v}_x \vec{i} + \vec{v}_y \vec{j} + \vec{v}_z \vec{k}

Exercises

Exercise 1

Verify the distributive law of vector operations: 2(a⃗+b⃗)=2a⃗+2b⃗2(\vec{a} + \vec{b}) = 2\vec{a} + 2\vec{b}, where a⃗=(1,2,3)\vec{a} = (1, 2, 3) and b⃗=(4,5,6)\vec{b} = (4, 5, 6).

properties of operationsthe distributive lawverifying a vector identity

Approach:

Compute the two sides of the equality separately and check whether they are equal.

Steps:

  1. The left-hand side: 2(a⃗+b⃗)=2((1,2,3)+(4,5,6))=2(5,7,9)=(10,14,18)2(\vec{a} + \vec{b}) = 2((1,2,3) + (4,5,6)) = 2(5,7,9) = (10,14,18)

  2. The right-hand side: 2a⃗+2b⃗=2(1,2,3)+2(4,5,6)=(2,4,6)+(8,10,12)=(10,14,18)2\vec{a} + 2\vec{b} = 2(1,2,3) + 2(4,5,6) = (2,4,6) + (8,10,12) = (10,14,18)

  3. The two sides are equal, so the distributive law holds

Answer: the distributive law is verified.

Exercise 2

Prove the associative law of vector addition: (a⃗+b⃗)+c⃗=a⃗+(b⃗+c⃗)(\vec{a} + \vec{b}) + \vec{c} = \vec{a} + (\vec{b} + \vec{c}).

properties of operationsthe associative lawproving a vector identity

Approach:

Use the coordinate representation of vectors to prove it.

Steps:

  1. The left-hand side: (a⃗+b⃗)+c⃗=((ax+bx,ay+by)+(cx,cy))=(ax+bx+cx,ay+by+cy)(\vec{a} + \vec{b}) + \vec{c} = ((a_x+b_x, a_y+b_y) + (c_x, c_y)) = (a_x+b_x+c_x, a_y+b_y+c_y)

  2. The right-hand side: a⃗+(b⃗+c⃗)=((ax,ay)+(bx+cx,by+cy))=(ax+bx+cx,ay+by+cy)\vec{a} + (\vec{b} + \vec{c}) = ((a_x, a_y) + (b_x+c_x, b_y+c_y)) = (a_x+b_x+c_x, a_y+b_y+c_y)

  3. The corresponding coordinates of the two sides are equal, so the associative law holds

Answer: the associative law is proved.

Summary

Symbols Used in This Article

SymbolTypePronunciation/ExplanationMeaning in This Article
a⃗+b⃗\vec{a} + \vec{b}Vector operationvector additionVector addition
ka⃗k\vec{a}Scalar multiplicationscalar multiplicationScalar multiplication
0⃗\vec{0}Vectorzero vectorThe zero vector
−a⃗-\vec{a}Negative vectornegative vectorThe negative vector
F⃗total\vec{F}_{total}VectorF totalThe resultant force
s⃗total\vec{s}_{total}Vectors totalThe total displacement
r⃗\vec{r}VectorrThe position along the ray
o⃗\vec{o}VectoroThe position of the light source
d⃗\vec{d}VectordThe direction of the ray
v⃗\vec{v}VectorvThe velocity vector
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