Inner product of two vectors 

Inner product of two vectors \(x\) and \( y\) denoted by \((x,y)\), is defined as: \[ (x,y)=x^Ty. \] For example the inner product of the two vectors \( x= \begin{bmatrix} 3 \\ 2 \\ 1 \\ 1 \end{bmatrix} \) and \( y= \begin{bmatrix} 1 \\ 2 \\ 3 \\ 4 \end{bmatrix} \) is calculated as: \[   (x,y)=x^Ty= \left[ \begin{array}{cccc} 3&2&1&1\\ \end{array} \right] \left[ \begin{array}{c} 1 \\ 2 \\ 3 \\ 4 \end{array} \right] =(3\times 1)+(2\times 2)+(1\times 3) +(1\times 4)=14.  \]

Norm or length of a vector

The length or norm of a vector \(x\), denoted as \(\|x\|\), is defined as:

\[ \| x \| = \sqrt{(x, x)}= \sqrt{x^Tx}. \] For example, the length of the vector \( x= \begin{bmatrix} 3 \\ 2 \\ 1 \\ 1 \end{bmatrix} \) is computed as: \[ \| x \|=\sqrt{ (3\times 3)+ (2\times 2)+ (1\times 1)+(1\times 1)}=\sqrt{15} \] 

Distance between two vectors

The distance between two vectors \(x\) and \(y\) is defined as the norm of \(u-v\), and is given as \( \|u-v\|\).

Angle between two vectors and orthogonality

The angle between two vectors \(x\) and \(y\)

is defined to be the number \(\theta \in [0, \pi] \)  such that:

\[\cos (\theta) = \frac{ (x,y)}{ \|x\| \|y\|}.\] 

Two vectors are said to be orthogonal (or normal to each other) if the angle between them is \(\pi/2\), or in other words when their inner product equals zero: \( (x,y)=x^Ty=0 \) .