A locational adposition places the located object in space relative to the reference object. However, instead of placing the located object in a specific position, the adposition places it in an area. The PP achter de camerabehind the camera in (3a) denotes the entire gray area in Figure 1. This is clear from the fact that the located object need not occupy a specific point in space, as in (3a), but can also be more scattered, as in (3b); cf. Section 32.3.1.1, sub III, for a more detailed discussion.

The area denoted by the adpositional phrase can be restricted by the use of modifiers. For example, the use of the modifier rechtstraight in (4a) restricts the area in such a way that only the noun phrases de fotograaf and lamp 3 can be substituted for “NP”. The use of the modifier vlakright in (4b) restricts the area so that, under the right pragmatic conditions, only the noun phrase de fotograaf can be substituted for “NP”.

The difference between modifiers like rechtstraight and vlakright is that the former specifies the spatial relation between the located and the reference object with respect to the orientation, while the latter specifies the distance between the two. The fact that these two kinds of modifications are possible is not accidental; Section 32.3.1.1 suggested that locational PPs, instead of referring to an area, can be considered to denote sets of vectors that take the reference object as their starting point; this is shown in the more abstract representation of the situation in Figure 1 given in Figure 2, which is also taken from Section 32.3.1.1, sub III.

The PP achter de camera refers to the set of vectors V‹x,y› that originate in (0,0) (viz. the position of the camera) and have their endpoint somewhere in the gray area, i.e. for which y > 0. Modifiers are used to restrict this set. The modifiers of orientation select a subset based on the direction of the vectors: the modifier rechtstraight in (4a) selects the subset of vectors for which x = 0, while the modifier schuindiagonally would select the vectors for which x ≠ 0. The modifiers of distance select a subset based on the magnitude (length) of the vector: the modifier vlakright in (4b) restricts the set of vectors to those that are smaller than some contextually given magnitude m, while the modifier verfar restricts the set to those vectors that are longer than some contextually given magnitude m', where m < m'.

In the examples above, the reference object is represented as a point in space. The vector approach to prepositions becomes more complicated when the reference object is represented as a three-dimensional object. Consider the examples in (5).

Example (5a) can refer to the situation in Figure 3A if the nominal modifier is twee kilometer but not if it is vier kilometer. Similarly, (5b) can refer to Figure 3B if the adjectival modifier is recht, but not if it is schuin. The contrasts found in (5) would of course be unexpected if the vectors and were both part of the vector set denoted by the adpositional phrases. This suggests that some minimality requirement is involved: only the shortest vectors going from the reference object to the located object are part of the set denoted by the adpositional phrase.

Note that the modifier recht in (5b) can also be used to express that the lamp hangs right over the center of the table. In this reading, the table may be mentally represented as a point in space, but we will not dwell on this issue.

Now that we have recapitulated the basic aspects of vector theory, we can discuss the modification of spatial adpositional phrases in more detail. Section 34.1.2 will show that vector theory provides us with a proper understanding of the modification possibilities: PPs headed by prepositions denoting sets of vectors can be modified for orientation and distance, which is clearly related to the fact that vectors have an orientation and a magnitude. Given the fact that the null vector lacks these properties (the magnitude of the null vector is zero, so the vector has no orientation either), it is to be expected that locational PPs denoting the null vector cannot be modified by orientation or distance modifiers. Section 34.1.3 will show that this prediction is indeed borne out, despite seeming counterexamples. Finally, Section 34.1.4 discusses the modification of directional adpositional phrases; in this case, the possibilities are limited to modifiers expressing the extent of the implied path actually traversed.