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

No, its length is

( 1^{2}+ 1^{2}+ 1^{2}) = 4 = 2

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 the inverse of its length:**v**/**| v |**

Often this idea is written as a formula (the little subscript "u" is supposed to mean "unit vector"):

v=_{u}v/| v |

If **v** = (x, y, z)^{T} then:

v=_{u}v/| v |= ( x /| v |, y/| v |, z/| v |)^{T}