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

No, its length

√( 1^{2}+ 1^{2}+ 1^{2}) = √3 = 1.7320508

is not one.

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 ofv,| v |. Scalevby dividing by its length:v/| v |

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

v=_{u}v/|v|

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

v=_{u}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}.