A vector is an ordered list of numbers: (3, −1, 2) has three components and lives in a three-dimensional space. Order matters — (3, −1, 2) and (2, −1, 3) are different vectors — and each position works as a coordinate axis. It can be read as a point (its coordinates) or as an arrow from the origin to that point; both readings are useful (Chollet, 2021, sec. 2.3.5; M. Mitchell, 2019, ch. 11).
Tiny example. With the vocabulary (the, cat, dog, eats), the sentence “the cat eats” is represented as (1, 1, 0, 1): a 1 for each word present, a 0 for each word absent. This is the bag-of-words representation, which already turns texts into points in a space (Alammar & Grootendorst, 2024, p. 7).
In machine learning almost everything ends up as a vector: an image, a sound, a word. Embeddings are learned vectors with hundreds or thousands of components; the text models Alammar and Grootendorst show use 384 or 768 (2024, pp. 60–62). Beware of the word “dimension”: a 5D vector has a single axis with five entries, whereas a 5D tensor has five axes (Chollet, 2021, sec. 2.2); see matrix and tensor.
For this audience there is a familiar bridge: conceptual or similarity spaces, in which each dimension is a discriminable quality and closeness between vectors measures resemblance (Gärdenfors, 2000; Buckner, 2024, p. 114). The disanalogy is that in learned vectors the dimensions almost never correspond to humanly identifiable properties: picturing one axis for “human” and another for “newborn” is a teaching simplification, not a description (Alammar & Grootendorst, 2024, p. 9). What turns this geometry into comparisons of meaning are operations such as the dot product and cosine similarity; what makes it strange is high dimensionality.