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) .
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.
Normalize the vector .
The norm of is:
The unit vector u with the same direction as v will be:
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
Calculate the distance between the vectors and in .
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