Translating a Vector

Translating a vector is quite different than rotating a vector. For translation to occur, the transformation matrix must be altered with an additional row and column. For example, An identity matrix allowing translation is represented by a 4x4 matrix as follows:

This also implies that for a vector to be rotated/translated, the vector must also contain an additional component as follows:

The fourth component is sometimes refer to as the w component and must be set to 1.

Doing so, the vector will be able to be translated or rotated. For example, let's rotate vector u 90 degrees counterclockwise about the x-axis and translate it to point (2,1,3)

Hope it helps.