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Unit Vectors 

Unit vectors are defined as vectors whose length is exactly one. This means you can use a unit vector to define direction and then assign a length to the vector to get a unique vector. 
The Standard Unit Vectors 

There are three standard unit vectors listed in the table below and shown in the figure.
source Wikipedia 

The Three Standard Unit Vectors  

Vector 
Definition 
\(\hat{i}\) 
the unit vector in the direction of the xaxis 
\(\hat{j}\) 
the unit vector in the direction of the yaxis 
\(\hat{k}\) 
the unit vector in the direction of the zaxis 
Every vector in 3dim space can be described as a linear combination of these three standard unit vectors. You will almost always see the same letters ('i', 'j' and 'k') used to indicate the standard unit vectors. However, you may see them written either as bold letters or with an 'arrow' instead of 'hat', i.e. \( \hat{i} = \vec{i} \).
Note  We recently ran across a book that used a little different notation. This book used \(\hat{x}\), \(\hat{y}\) and \(\hat{z}\). Also, some books use the 'hat' notation to always indicate unit vectors.
You now have two ways to describe a vector, \( \vec{v} = \langle v_1, v_2, v_3 \rangle \) or the same vector can be written \( \vec{v} = v_1 \hat{i} + v_2 \hat{j} + v_3 \hat{k} \).
Okay, so let's watch a video explaining this in more detail. He goes through an example and explains this whole idea of the standard unit vectors as he works it.
Khan Academy  standard unit vectors  
Finding a Unit Vector in a Certain Direction 

If you have a vector, let's call it \(\vec{v}\), and you want to find a unit vector in the same direction as \(\vec{v}\), you just multiply the vector by the scalar \( 1/ \norm{\vec{v}} \). Said another way, you divide the vector by its length. This gives you the unit vector, \(\vec{u}\), in the same direction as vector \(\vec{v}\).
This is written as \(\displaystyle{ \vec{u} = \frac{\vec{v}}{\norm{\vec{v}}} }\).
You will often see a \(\vec{u}\) used to define a unit vector but not always. So make sure to check the context to determine exactly what is meant and keep track of unit vectors. They are important to vector calculus.
Okay, time for some practice problems. 
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Practice Problems 

Level A  Basic 
Practice A01  

Find the unit vector in the direction of \(\langle 3,4 \rangle\).  
solution 
Practice A02  

Find the unit vector in the direction of \(\vec{w}=\langle 1/2,1/8 \rangle\).  
solution 
Practice A03  

Find the unit vector in the direction of \(\vec{w}=\langle 0,5 \rangle\).  
solution 
Practice A04  

Calculate \(\vec{a}+\vec{b}\) for \(\vec{a}=3\hat{i}+2\hat{j} \), \( \vec{b}=2\hat{i}+4\hat{j}\).  
solution 