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Does the following represent a unit vector: (1, 1, 1)T ?


No, its length

( 12 + 12 + 12 ) =  3 =  1.7320508

is not one.

Creating Unit Vectors

Unit vectors have a length of one. If you have a particular vector v you can use it to make a unit vector. This is called normalizing the vector:

Calculate the length of v,   | v |. 
Scale v by dividing by its length:    v / | v |  

Often this idea is written as a formula. The subscript "u" means "unit vector".

vu = v / |v|

If v   =   (x, y, z)T   then:

vu  =  v / |v|  =  ( x /|v|,  y/|v|,  z/|v| )T

Different books use different notation for this. Sometimes a "hat" is used for unit vectors:   î   but this does not show up well in browsers. Sometimes a unit vector is called a "normalized" vector.


Normalize (3, 4)T.