# The Norm Of A Vector

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 .

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.

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 .