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In mathematics, the Euclidean distance between two points in Euclidean space is the length of the line segment between them. It can be calculated from the Cartesian coordinates of the points using the Pythagorean theorem , and therefore is occasionally called the Pythagorean distance .
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", as in ^ (pronounced "v-hat"). The term normalized vector is sometimes used as a synonym for unit vector. The normalized vector û of a non-zero vector u is the ...
where θ is the measure of the angle between a and b, and n is a unit vector perpendicular to both a and b which completes a right-handed system. The right-handedness constraint is necessary because there exist two unit vectors that are perpendicular to both a and b, namely, n and (−n). An illustration of the cross product
Also, let Q = (x 1, y 1) be any point on this line and n the vector (a, b) starting at point Q. The vector n is perpendicular to the line, and the distance d from point P to the line is equal to the length of the orthogonal projection of on n. The length of this projection is given by:
Points in the polar coordinate system with pole O and polar axis L. In green, the point with radial coordinate 3 and angular coordinate 60 degrees or (3, 60°). In blue, the point (4, 210°). In mathematics, the polar coordinate system specifies a given point in a plane by using a distance and an angle as its two coordinates. These are
In differential geometry, the Gauss map of a surface is a function that maps each point in the surface to a unit vector that is orthogonal to the surface at that point. Namely, given a surface X in Euclidean space R 3 , the Gauss map is a map N : X → S 2 (where S 2 is the unit sphere ) such that for each p in X , the function value N ( p ) is ...
Here α, β, γ are the direction cosines and the Cartesian coordinates of the unit vector | |, and a, b, c are the direction angles of the vector v. The direction angles a , b , c are acute or obtuse angles , i.e., 0 ≤ a ≤ π , 0 ≤ b ≤ π and 0 ≤ c ≤ π , and they denote the angles formed between v and the unit basis vectors e x ...
The two polar coordinates of a point in a plane may be considered as a two dimensional vector. Such a vector consists of a magnitude (or length) and a direction (or angle). The magnitude, typically represented as r , is the distance from a starting point, the origin , to the point which is represented.