Theory

Vector space - addition and scaling

A vector is an element in a vector space, which is a collection of objects supporting addition and scalar coefficients.

Vector space operations

• $\mathbf{v}+\mathbf{w}$ is a vector when $\mathbf{v},\mathbf{w}=\text{vectors}$
• $\lambda \mathbf{v}$ is a vector when $\mathbf{v}=\text{vector},\lambda =\text{scalar}$

Properties of vector operations

• Commutativity: $\mathbf{v}+\mathbf{w}=\mathbf{w}+\mathbf{v}$
• Associativity: $\mathbf{u}+(\mathbf{v}+\mathbf{w})=(\mathbf{u}+\mathbf{v})+\mathbf{w}$
• Distributivity: $\lambda (\mathbf{v}+\mathbf{w})=\lambda \mathbf{v}+\lambda \mathbf{w}$
• Zero vector: $\mathbf{v}+0=\mathbf{v}$

The elements $\lambda \mathbf{v}$ also satisfy obvious properties like:
$1\mathbf{v}=\mathbf{v}$ and $(\lambda +\mu )\mathbf{v}=\lambda \mathbf{v}+\mu \mathbf{v}$ and $\lambda (\mu \mathbf{v})=(\lambda \mu )\mathbf{v}$

Scalars
Scalars (“scal-ers”) could be taken in any field, but for this class, they are just real numbers $\lambda \in \mathbb{R}$.

Components of vectors

In this class we consider vectors in the plane ${\mathbb{R}}^2$ and vectors in space ${\mathbb{R}}^3$.

Vectors in the plane are ordered pairs $\mathbf{v}=(a_1,a_2)$ with $a_1,a_2\in \mathbb{R}$.
Vectors in space are ordered triples $\mathbf{v}=(a_1,a_2,a_3)$ with $a_1,a_2,a_3\in \mathbb{R}$.

Addition is componentwise: $(a_1,a_2)+(b_1,b_2)=(a_1+b_1,a_2+b_2)$
Scalars act by components too: $\lambda (a_1,a_2)=(\lambda a_1,\lambda a_2)$
(Similarly for vectors in space.)

Standard basis vectors
These special vectors have their own symbols:
${\mathbf{e}}_{\mathbf{i}}=(1,0,0),{\mathbf{e}}_{\mathbf{j}}=(0,1,0),{\mathbf{e}}_{\mathbf{k}}=(0,0,1)$

Components
Most often the word ‘component’ of a vector refers to one of the numbers $a_i$ in the ordered sequence $(a_1,a_2)$ or $(a_1,a_2,a_3)$.

Sometimes ‘component’ refers to the basis vector scaled by this number, e.g. $a_1{\mathbf{e}}_{\mathbf{i}}$.
Notice that we always have $(a_1,a_2,a_3)=a_1{\mathbf{e}}_{\mathbf{i}}+a_2{\mathbf{e}}_{\mathbf{j}}+a_3{\mathbf{e}}_{\mathbf{k}}$, which is the reason ‘component’ is sometimes used in this second way.

Other bases
It is possible to select a different set of special vectors, e.g. $\mathbf{B}=\{{\mathbf{u}}_1,{\mathbf{u}}_2,{\mathbf{u}}_3\}$, having the property that any vector $\mathbf{v}$ can be written uniquely as a linear combination in the same way: $\mathbf{v}=a_1{\mathbf{u}}_1+a_2{\mathbf{u}}_2+a_3{\mathbf{u}}_3$. Any such set is called a basis, and the $a_i$ in this case are called components in the basis $\mathbf{B}$. This topic is considered further in Linear Algebra.

Coordinates
Mathematicians say ‘components’ for the parts of a vector, and ‘coordinates’ for the quantities that are used to identify points of space. Since the spaces ${\mathbb{R}}^2$ and ${\mathbb{R}}^3$ are vector spaces, the tuples $(a_1,a_2)$ or $(a_1,a_2,a_3)$ actually give coordinates and vector components!

In more general curved spaces, points cannot be “added together,” so this relation between coordinates and components breaks down.

Geometric vectors

It is possible to consider vectors purely geometrically as the combined data of direction and magnitude in space.
(Provided one has defined space purely geometrically, which nobody does.)