This post will be about the norm of a vector. It is assumed that the reader knows about vectors where a vector in is of the form .

**Definition Of A Norm**

The norm of a vector in is defined as:

Sometimes the norm of a vector is referred as the length of or the magnitude of

In three dimensions, a vector in is with the norm as:

In two dimensions, we have a vector in . Its norm is:

In one dimension, we have a the vector is just on the real line . The absolute value is a special case of the norm and it is expressed as:

Note that we square the number to ensure a positive number and then take the square root. Doing this is the same as taking the absolute value.

The norm is no longer a vector as it is a scalar/number (with no direction).

**Some Properties Of The Norm**

Here are some properties of a vector in with a scalar (real number) .

**Unit Vectors**

A vector of a norm of 1 is a unit vector. Unit vectors are of use when length is not relevant. The unit vector is defined as:

where **v** is a non-zero vector in .

When we obtain a unit vector **u** from **v**, it is called normalizing **v**.

__Example One__

Normalize the vector .

Answer:

The norm of is:

The unit vector **u** with the same direction as **v** will be:

__Example Two__

Given the vector . Find the unit vector **u** such that it has the same direction as **v**.

Answer:

The norm of is:

Our unit vector **u** will be:

**Standard Unit Vectors**

You may encounter standard unit vectors (of norm 1) in the form of:

in . For , you may see:

.

For example, we can express the vector (2, 1) as . Likewise, the vector (-3, -1, 5) can be expressed as .

In the general case in , the standard unit vectors would be:

and any vector can be expressed as a linear combination as follows:

**Distance Between Two Vectors**

Recall that the distance between points and in 2-space is:

In a three-dimensional setting, the distance between points and is:

We can now extend this to n-th dimensional space .

In , the distance between vectors and is

*Example*

Calculate the distance between the vectors and in .

Answer:

We apply the distance formula.

Reference: Elementary Linear Algebra (10th Editon) by Howard Anton

The image is taken from https://www.mathsisfun.com/algebra/images/vector-mag-dir.gif