Vectors

Abigail Huettig

29 Vectors

Vectors in two-dimensions are made up of an initial point and a terminal point. In the $xy$ -coordinate plane, the initial point is at (0,0), and the terminal point is at some other coordinate $(x,y)$ [16]. If we have a vector

$\textbf{v}=(x_1,x_2),$

then ( $x_1,x_2)$ is the terminal point, and $x_1$ and $x_2$ are the components [16]. One more important fact is that a vector is denoted with a bold lowercase letter [16]. Vectors do not have to be two-dimensional. In fact, they can have as many dimensions as we want. A vector in $n$ -dimensions is denoted as $\textbf{v}=(x_1, x_2, ..., x_n)$ [16].

Two basic operations can be performed on vectors, vector addition and scalar multiplication. Vector addition is done component-wise. That is, for two vectors $\textbf{u}=(x_1,x_2,...,x_n)$ and $\textbf{v}=(y_1, y_2, ..., y_n)$ in $n$ dimensions,

$\textbf{u}+\textbf{v}=(x_1+y_1,x_2+y_2,...,x_n+y_n).[16]$

For a vector $\textbf{u}=(x_1,x_2,...,x_n)$ and scalar $c$ , scalar multiplication is defined as

$c\textbf{u}=(cx_1,cx_2,...,cx_n). [16]$

As you can see, these operations are performed in much the same way as matrix addition and scalar multiplication of matrices. This makes sense as matrices are a subset of vectors.

We have a great amount of information to discuss relating to vectors so let’s begin our next section on vector spaces.

29 Vectors

License

Share This Book