Ihat

Ihat, or i-hat, is the standard symbol used to denote the unit vector along the x-axis in a Cartesian coordinate system. It forms part of the right-handed basis i-hat, j-hat, k-hat, where i-hat points in the positive x-direction, j-hat in the positive y-direction, and k-hat in the positive z-direction. A vector can be expressed as v = v_x i-hat + v_y j-hat + v_z k-hat, with i-hat having magnitude 1 and being orthogonal to both j-hat and k-hat.

Notation and conventions vary slightly. In mathematical notation, i-hat is often written as î, or as \hat{i} in LaTeX. In plain text, ihat or i-hat is commonly used. The unit vectors form an orthonormal basis, so i-hat · i-hat = 1, i-hat · j-hat = 0, and i-hat · k-hat = 0.

Usage and applications include expressing vectors, performing dot and cross products, and describing physical fields such as velocity, force, and electric fields. In multivariable calculus, the gradient operator can be written as ∇f = ∂f/∂x i-hat + ∂f/∂y j-hat + ∂f/∂z k-hat. In physics, engineering, and computer graphics, i-hat serves as a fundamental directional component for decomposing and manipulating spatial quantities.

Other uses of the term “hat” appear in different contexts, notably in statistics and machine learning where hats denote estimators (for example, θ-hat). Those uses are typically separate from the vector i-hat notation.

See also: j-hat, k-hat, Cartesian coordinate system, vector notation, unit vector.