Unit vector

In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, a unit vector in a <a href="/facts/Normed_vector_space/rmDljfyE">normed vector space</a> is a <a href="/facts/Vector_(mathematics_and_physics)/U8mM7A5F">vector</a> (often a <a href="/facts/Vector_(geometry)/bCLrdksy">spatial vector</a>) of <a href="/facts/Norm_(mathematics)/xIbR4uE1">length</a> 1. A unit vector is often denoted by a lowercase letter with a <a href="/facts/Circumflex/kg8xlo6W">circumflex</a>, or "hat", as in 
 
 
 
 
 
 
 
 v
 
 ^
 
 
 
 
 
 {\displaystyle {\hat {\mathbf {v} }}}
 
 (pronounced "v-hat"). The term normalized vector is sometimes used as a synonym for unit vector.
The normalized vector û of a non-zero vector u is the unit vector in the direction of u, i.e.,

u
              ^
            
          
        
        =
        
          
            
              u
            
            
              ‖
              
                u
              
              ‖
            
          
        
        =
        (
        
          
            
              u
              
                1
              
            
            
              ‖
              
                u
              
              ‖
            
          
        
        ,
        
          
            
              u
              
                2
              
            
            
              ‖
              
                u
              
              ‖
            
          
        
        ,
        .
        .
        .
        ,
        
          
            
              u
              
                n
              
            
            
              ‖
              
                u
              
              ‖
            
          
        
        )
      
    
    {\displaystyle \mathbf {\hat {u}} ={\frac {\mathbf {u} }{\|\mathbf {u} \|}}=({\frac {u_{1}}{\|\mathbf {u} \|}},{\frac {u_{2}}{\|\mathbf {u} \|}},...,{\frac {u_{n}}{\|\mathbf {u} \|}})}

where ‖u‖ is the <a href="/facts/Norm_(mathematics)/xIbR4uE1">norm</a> (or length) of u and 
 
 
 
 ‖
 
 u
 
 ‖
 =
 (
 
 u
 
 1
 
 
 ,
 
 u
 
 2
 
 
 ,
 .
 .
 .
 ,
 
 u
 
 n
 
 
 )
 
 
 {\textstyle \|\mathbf {u} \|=(u_{1},u_{2},...,u_{n})}
 
. 
The proof is the following: 
 
 
 
 ‖
 
 
 
 u
 ^
 
 
 
 ‖
 =
 
 
 
 
 
 
 u
 
 1
 
 
 
 
 u
 
 1
 
 
 2
 
 
 +
 .
 .
 .
 +
 
 u
 
 n
 
 
 2
 
 
 
 
 
 
 2
 
 
 +
 .
 .
 .
 +
 
 
 
 
 u
 
 n
 
 
 
 
 u
 
 1
 
 
 2
 
 
 +
 .
 .
 .
 +
 
 u
 
 n
 
 
 2
 
 
 
 
 
 
 2
 
 
 
 
 =
 
 
 
 
 
 u
 
 1
 
 
 2
 
 
 +
 .
 .
 .
 +
 
 u
 
 n
 
 
 2
 
 
 
 
 
 u
 
 1
 
 
 2
 
 
 +
 .
 .
 .
 +
 
 u
 
 n
 
 
 2
 
 
 
 
 
 
 =
 
 
 1
 
 
 =
 1
 
 
 {\textstyle \|\mathbf {\hat {u}} \|={\sqrt {{\frac {u_{1}}{\sqrt {u_{1}^{2}+...+u_{n}^{2}}}}^{2}+...+{\frac {u_{n}}{\sqrt {u_{1}^{2}+...+u_{n}^{2}}}}^{2}}}={\sqrt {\frac {u_{1}^{2}+...+u_{n}^{2}}{u_{1}^{2}+...+u_{n}^{2}}}}={\sqrt {1}}=1}

A unit vector is often used to represent <a href="/facts/Direction_(geometry)/Tj24mTJ8">directions</a>, such as <a href="/facts/Normal_direction/jhtLIhcu">normal directions</a>.
Unit vectors are often chosen to form the <a href="/facts/Basis_(linear_algebra)/89IPoN6c">basis</a> of a vector space, and every vector in the space may be written as a <a href="/facts/Linear_combination/CVjL6kuN">linear combination</a> form of unit vectors.

Unit vector open-in-new

Unit vector