A First Course in Linear Algebra, 2017a

4.4. Length of a Vector 155
We can now use our understanding of the distance between two points to define what is meant by the
length of a vector. Consider the following definition.
Definition 4.14: Length of a Vector
Let ⃗u =[u 1 ···u n ] T beavectorinR n . Then, the length of ⃗u, written ‖⃗u‖ is given by
√
‖⃗u‖ = u 2 1 + ···+ u2 n
This definition corresponds to Definition 4.10, if you consider the vector ⃗u to have its tail at the point
0 =(0,···,0) and its tip at the point U =(u 1 ,···,u n ). Then the length of⃗u is equal to the distance between
0andU, d(0,U). In general, d(P,Q)=‖ −→ PQ‖.
Consider Example 4.11. By Definition 4.14, we could also find the distance between P and Q as the
length of the vector connecting them. Hence, if we were to draw a vector −→ PQ with its tail at P and its point
at Q, this vector would have length equal to √ 47.
We conclude this section with a new definition for the special case of vectors of length 1.
Definition 4.15: Unit Vector
Let ⃗u be a vector in R n . Then, we call ⃗u a unit vector if it has length 1, that is if
‖⃗u‖ = 1
Let ⃗v be a vector in R n . Then, the vector ⃗u which has the same direction as ⃗v but length equal to 1 is
the corresponding unit vector of⃗v. This vector is given by
⃗u = 1
‖⃗v‖ ⃗v
We often use the term normalize to refer to this process. When we normalize avector,wefindthe
corresponding unit vector of length 1. Consider the following example.
Example 4.16: Finding a Unit Vector
Let⃗v be given by
⃗v = [ 1 −3 4 ] T
Find the unit vector ⃗u which has the same direction as⃗v .
Solution. We will use Definition 4.15 to solve this. Therefore, we need to find the length of ⃗v which, by
Definition 4.14 is given by
√
‖⃗v‖ = v 2 1 + v2 2 + v2 3
Using the corresponding values we find that
‖⃗v‖ =
√
1 2 +(−3) 2 + 4 2