Free and Localized Vectors
A free vector( or non-localized vector) is a vector of which only the magnitude and direction are specified, not the position or line of action. Displacing it parallel to itself leaves it unchanged.
A localized vector is a vector where line of action and position are as important as magnitude and direction. These vectors change with change in position and direction.
Equality of Vectors
Two vectors are said to be equal only when they have same direction and magnitude. For example, two cars travelling with same speed in same direction. If they are travelling in opposite directions with same speed, then the vectors are unequal.