vectorcomponent

A vector component is a measure of how much a vector lies in a given direction. It is commonly discussed in two related forms: the scalar component, which is the magnitude of the projection of the vector onto a direction, and the vector component, which is the projection vector itself in that direction.

Mathematically, let v be a vector and a be a direction vector. If a_hat is a unit vector in the direction of a, then:

- The scalar component of v along a is the scalar projection v · a_hat, equal to (v · a) / ||a||.

- The vector component (or projection) of v along a is the projection vector proj_a(v) = (v · a_hat) a_hat = ((v · a) / ||a||^2) a.

In any orthonormal basis, the components of v along each basis vector e_i are simply the dot products v · e_i, which reproduce v as a sum of these components times the basis vectors: v = sum_i (v · e_i) e_i. In standard Cartesian coordinates, these components are the coordinates x, y, z of v.

Examples help illustrate the idea. If v = (3, 4) in the plane and we project onto the x-axis (unit vector (1, 0)), the scalar component is 3 and the vector component is (3, 0). Projecting onto the direction (1, 1)/√2 yields a scalar component v · a_hat = 7/√2 and a vector component proj_a(v) = (7/2, 7/2).

Applications include resolving forces into components, analyzing motion, and performing coordinate transforms. For non-orthonormal bases, components are obtained by solving the corresponding linear system; the concept remains the projection of v onto the chosen direction or basis.