# How do you determine whether a vector is orthogonal, parallel, or neither?

Oct 30, 2015

Of course you can check whether a vector is orthogonal, parallel, or neither with respect to some other vector. So, let's say that our vectors have $n$ coordinates.

The concept of parallelism is equivalent to the one of multiple, so two vectors are parallel if you can obtain one from the other via multiplications by a number: for example, $v = \left(3 , 2 , - 5\right)$ is parallel to $w = \left(30 , 20 , - 50\right)$ and to $z = \left(- 3 , - 2 , 5\right)$, because $w = 10 \cdot v$, and $z = \left(- 1\right) \cdot v$.

To check if two vectors are orthogonal, instead, you can use the scalar product. If you have two vectors
$a = \left({a}_{1} , \ldots , {a}_{n}\right)$ and $b = \left({b}_{1} , \ldots , {b}_{n}\right)$, the scalar product $a \cdot b$ is defined (for numerical vectors) as

$a \cdot b = {a}_{1} {b}_{1} + {a}_{2} {b}_{2} + \ldots + {a}_{n} {b}_{n} = {\sum}_{i = 1}^{n} {a}_{i} {b}_{i}$

The scalar product is often used to define the concept of orthogonality itself, when working with non-numerical vectors, which you can't properly visualize, and two vectors are said to be orthogonal if their scalar product is zero. For example, if you consider the vectorial space of continuous function, how can you "see" if two functions are orthogonal? You define a proper scalar product on that space, and if $f \cdot g = 0$, then $f$ and $g$ are orthogonal.

Numerical examples of orthogonal vectors can be

$a = \left(3 , 2 , 1\right)$, $b = \left(1 , 1 , - 6\right)$, since

$a \cdot b = 3 \cdot 1 + 2 \cdot 1 + 1 \cdot \left(- 6\right) = 6 - 6 = 0$.

or for example an easy check that the $x$ and $y$-axis are orthogonal (of course)! is

$x = \left(1 , 0\right)$, $y = \left(0 , 1\right)$, and

$x \cdot y = 1 \cdot 0 + 0 \cdot 1 = 0 + 0 = 0$.