Vectors and Vector Spaces

Objects You Can Add and Scale

A vector is an object that has both magnitude and direction — a displacement, a velocity, a force. In coordinates, a vector in the plane is a pair $(v_1, v_2)$ and in $n$ dimensions an $n$-tuple. But the deeper idea is structural: a vector space is any set of objects that can be added together and scaled by numbers (scalars), obeying a short list of natural rules.

Two operations define a vector space:

• Vector addition — add componentwise: $(a_1, a_2) + (b_1, b_2) = (a_1 + b_1, a_2 + b_2)$. Geometrically, place them tip-to-tail.
• Scalar multiplication — stretch or shrink: $c(v_1, v_2) = (c v_1, c v_2)$. A negative scalar also reverses direction.

These obey the vector-space axioms: addition is commutative and associative, there is a zero vector $\mathbf{0}$ (adding it changes nothing), every vector has a negative, and scalar multiplication distributes over addition. Any set satisfying these axioms is a vector space — and remarkably, this includes not just arrows but polynomials, matrices, and functions, which is why the abstraction is so powerful.

The magnitude (norm) of a vector measures its length: in the plane, $\|v\| = \sqrt{v_1^2 + v_2^2}$ by the Pythagorean theorem. A unit vector has norm 1 and captures pure direction. In computing, vectors are everywhere — coordinates in graphics, feature vectors in machine learning, embeddings in language models.

Common pitfall: treating vectors as mere lists of numbers and losing the operations, or confusing a vector's magnitude with its components. A vector is defined by how it adds and scales, not just its entries; and its magnitude is $\sqrt{v_1^2 + v_2^2}$ (a single length), not the individual components $v_1, v_2$. Adding vectors is componentwise — you do not add magnitudes (the length of a sum is generally not the sum of the lengths).

Two vectors from the origin joined tip-to-tail, the accent resultant drawn from start to final tip, with its Pythagorean length labeled — showing componentwise addition, not magnitude addition.

$$\|v\| = \sqrt{v_1^2 + v_2^2}$$