Unit vector

In mathematics, a unit vector in a normed vector space is a vector (often a spatial vector) of length 1. A unit vector is often denoted by a lowercase letter with a circumflex, or 'hat': ı ^ {displaystyle {hat {imath }}} (pronounced 'i-hat'). The term direction vector is used to describe a unit vector being used to represent spatial direction, and such quantities are commonly denoted as d. Two 2D direction vectors, d1 and d2 are illustrated. 2D spatial directions represented this way are numerically equivalent to points on the unit circle.A normal vector n ^ {displaystyle mathbf {hat {n}} } to the plane containing and defined by the radial position vector r r ^ {displaystyle rmathbf {hat {r}} } and angular tangential direction of rotation θ θ ^ {displaystyle heta {oldsymbol {hat { heta }}}} is necessary so that the vector equations of angular motion hold.In terms of polar coordinates; n ^ = r ^ × θ ^ {displaystyle mathbf {hat {n}} =mathbf {hat {r}} imes {oldsymbol {hat { heta }}}} One unit vector e ^ ∥ {displaystyle mathbf {hat {e}} _{parallel }} aligned parallel to a principal direction (red line), and a perpendicular unit vector e ^ ⊥ {displaystyle mathbf {hat {e}} _{ot }} is in any radial direction relative to the principal line.Unit vector at acute deviation angle φ (including 0 or π/2 rad) relative to a principal direction. In mathematics, a unit vector in a normed vector space is a vector (often a spatial vector) of length 1. A unit vector is often denoted by a lowercase letter with a circumflex, or 'hat': ı ^ {displaystyle {hat {imath }}} (pronounced 'i-hat'). The term direction vector is used to describe a unit vector being used to represent spatial direction, and such quantities are commonly denoted as d. Two 2D direction vectors, d1 and d2 are illustrated. 2D spatial directions represented this way are numerically equivalent to points on the unit circle. The same construct is used to specify spatial directions in 3D. As illustrated, each unique direction is equivalent numerically to a point on the unit sphere. The normalized vector or versor û of a non-zero vector u is the unit vector in the direction of u, i.e., where |u| is the norm (or length) of u. The term normalized vector is sometimes used as a synonym for unit vector. Unit vectors are often chosen to form the basis of a vector space. Every vector in the space may be written as a linear combination of unit vectors. By definition, in a Euclidean space the dot product of two unit vectors is a scalar value amounting to the cosine of the smaller subtended angle. In three-dimensional Euclidean space, the cross product of two arbitrary unit vectors is a third vector orthogonal to both of them having length equal to the sine of the smaller subtended angle. The normalized cross product corrects for this varying length, and yields the mutually orthogonal unit vector to the two inputs, applying the right-hand rule to resolve one of two possible directions. Unit vectors may be used to represent the axes of a Cartesian coordinate system. For instance, the unit vectors in the direction of the x, y, and z axes of a three dimensional Cartesian coordinate system are They are sometimes referred to as the versors of the coordinate system, and they form a set of mutually orthogonal unit vectors, typically referred to as a standard basis in linear algebra. They are often denoted using normal vector notation (e.g., i or ı → {displaystyle {vec {imath }}} ) rather than standard unit vector notation (e.g., ı ^ {displaystyle mathbf {hat {imath }} } ). In most contexts it can be assumed that i, j, and k, (or ı → , {displaystyle {vec {imath }},} ȷ → , {displaystyle {vec {jmath }},} and k → {displaystyle {vec {k}}} ) are versors of a 3-D Cartesian coordinate system. The notations ( x ^ , y ^ , z ^ ) {displaystyle (mathbf {hat {x}} ,mathbf {hat {y}} ,mathbf {hat {z}} )} , ( x ^ 1 , x ^ 2 , x ^ 3 ) {displaystyle (mathbf {hat {x}} _{1},mathbf {hat {x}} _{2},mathbf {hat {x}} _{3})} , ( e ^ x , e ^ y , e ^ z ) {displaystyle (mathbf {hat {e}} _{x},mathbf {hat {e}} _{y},mathbf {hat {e}} _{z})} , or ( e ^ 1 , e ^ 2 , e ^ 3 ) {displaystyle (mathbf {hat {e}} _{1},mathbf {hat {e}} _{2},mathbf {hat {e}} _{3})} , with or without hat, are also used, particularly in contexts where i, j, k might lead to confusion with another quantity (for instance with index symbols such as i, j, k, used to identify an element of a set or array or sequence of variables).