Vectors

A precise introduction to vectors, components, magnitude, direction, linear combinations, and geometric operations.

Orthogonal projection separates a vector into fitted and perpendicular components. A vector is decomposed into a component along a subspace and a right-angled residual. Basis arrows show how coordinates reconstruct the projected result.

Precise definition

A vector in $\mathbb R^n$ is an ordered tuple with componentwise addition and scalar multiplication. Geometrically it may represent displacement, velocity, or force, independent of its drawing location. The Euclidean norm is $\|v\|=\sqrt{v_1^2+\cdots+v_n^2}$, and a unit vector has norm 1.

Notation and mathematical language

The dot product is $u\cdot v=\sum u_iv_i=\|u\|\|v\|\cos\theta$. It measures alignment and defines orthogonality. A linear combination $au+bv$ combines directions with scalar weights. Components depend on the chosen coordinate axes and units.

Conceptual picture

Vector addition follows head-to-tail displacement; scalar multiplication changes magnitude and may reverse direction. The dot product converts two vectors to a scalar and connects algebraic components with angle and projection.

Fully worked example

Interpretation and application

Vectors model forces, navigation, data points, and states. A component can change under rotation of axes while the physical vector and its norm remain unchanged. Context determines whether Euclidean distance is meaningful for data.