A vector is a physical quantity representing both direction and magnitude.
Vectors can have an arbritrary number of dimensions.

Vectors are often written out as column vectors:

\[ \underline{v} = \begin{pmatrix} 1.3 \\ 2.4 \end{pmatrix} \]

Notation that may also indicate that a variable is a vector is: $\mathbf{v}$ or $\overrightarrow{v}$

Individual Elements

Vectors are indexed from 1.
\[ s_1 = 24 s_2 = 39 s_3 = 23 s_4 = 91 \\ \underline{s} = \begin{pmatrix} 24 \\ 39 \\ 23 \\ 91 \end{pmatrix} \]

Transposing

Transposing is the vector operation that reflects it over the main diagonal (runs from the top left to the bottom right).
The transpose of a vector is denoted by $\underline{V}^T$

\[ \underline{v} = \begin{pmatrix} 1.2 \\ 3.4 \\ 5.6 \end{pmatrix} \\ \\ \underline{v}^T = \begin{pmatrix} 1.2 & 3.4 & 5.6 \end{pmatrix} \]

Formal Definition

\[ {\underline{v}^T}_{ij} = \underline{v}_{ji} \]

Matrices can also be transposed.

Operations On Vectors

Addition

\[ \underline{w} = \underline{v} + \underline{u}\\ \\ \underline{w}_i = \underline{v}_i + \underline{u}_i \]
  • $\underline{u} + \underline{v}$ is a vector of the same dimension as $\underline{w}$
  • $\underline{u} + \underline{v} = \underline{v} + \underline{u}$
  • There is a unique vector known as $\underline{0}$ such that $\underline{v} + \underline{0} = \underline{v}$
  • There is always a vector $\underline{x}$ such that for any $\underline{v}$: $\underline{v} + \underline{x} = 0$
    • We call this vector $-\underline{v}$

Formally, addition in vectors behaves like addition in numbers.
Note that vectors of different dimension CANNOT be added.

Subtraction

\[ \underline{w} = \underline{v} - \underline{u}\\ \\ \underline{w}_i = \underline{v}_i - \underline{u}_i \]

Many of the same rules as addition apply for subtraction.

Scalar Product

The magnitude of a vector can be altered without changing its direction.
If it is multiplied by a negative scalar, however, its direction is inverted.

\[ \underline{i} = x\underline{v} \\ \underline{u}_i = x\underline{v}_i\\ \]

(We are assuming that for all $\underline{v}_i$ in $\underline{v}$ that $\underline{v}_i \in \mathbb{R}$)

  • If $\underline{v}$ is a vector and $x \in \mathbb{R}$ then $x\underline{v}$ is also a vector
  • $x(\underline{u} + \underline{v}) = x\underline{u} + x\underline{v}$
  • $(x + y)\underline{v} = x\underline{v} + y\underline{v}$
  • $x(y\underline{v}) = (xy)\underline{v}$
  • $1\underline{v} = \underline{v}$