vector components

An arrow drawn on paper is vivid but hard to compute with. Components are the trick that turns the arrow into numbers you can add, scale, and feed into a calculator. The idea: pick three perpendicular directions — the x, y and z axes — and report how far the vector reaches along each. Those three numbers ARE the vector, in the sense that they determine it completely.

Concretely, set up the standard basis arrows: i = (1, 0, 0) points one unit along x, j = (0, 1, 0) along y, k = (0, 0, 1) along z. Any space vector v can be reached by going a units in the i direction, then b in the j direction, then c in the k direction, so v = a i + b j + c k, written compactly as v = (a, b, c). Here a, b, c are the components (also called the scalar components). To find them from an arrow joining P(p_1, p_2, p_3) to Q(q_1, q_2, q_3), subtract tail from head: PQ = (q_1 - p_1, q_2 - p_2, q_3 - p_3). Once in components, the magnitude follows from the 3D Pythagorean theorem: |v| = sqrt(a^2 + b^2 + c^2).

Components are what make vector algebra mechanical: to add two vectors you add matching components; to scale by a number you multiply every component. Every formula later — dot product, cross product, the equation of a line — is just a recipe on these lists. One caveat to keep honest: the components depend on the axes you chose. Rotate the axes and the same physical arrow gets a different list of numbers, even though its true length and direction never moved. The vector is the invariant thing; its components are merely its address in a particular coordinate system.