## CGAL::Vector_2<Kernel>

### Definition

An object of the class Vector_2<Kernel> is a vector in the two-dimensional vector space  2. Geometrically spoken, a vector is the difference of two points p2, p1 and denotes the direction and the distance from p1 to p2.

CGAL defines a symbolic constant NULL_VECTOR. We will explicitly state where you can pass this constant as an argument instead of a vector initialized with zeros.

### Creation

 Vector_2 v ( Point_2 a, Point_2 b); introduces the vector b-a. Vector_2 v ( Segment_2 s); introduces the vector s.target()-s.source(). Vector_2 v ( Ray_2 r); introduces the vector having the same direction as r. Vector_2 v ( Line_2 l); introduces the vector having the same direction as l. Vector_2 v ( Null_vector NULL_VECTOR); introduces a null vector v. Vector_2 v ( Kernel::RT hx, Kernel::RT hy, Kernel::RT hw = RT(1)); introduces a vector v initialized to (hx/hw,hy/hw). Precondition: hw not equal to 0

### Operations

 bool v.operator== ( w) Test for equality: two vectors are equal, iff their x and y coordinates are equal. You can compare a vector with the NULL_VECTOR. bool v.operator!= ( w) Test for inequality. You can compare a vector with the NULL_VECTOR.

There are two sets of coordinate access functions, namely to the homogeneous and to the Cartesian coordinates. They can be used independently from the chosen kernel model.

 Kernel::RT v.hx () returns the homogeneous x coordinate. Kernel::RT v.hy () returns the homogeneous y coordinate. Kernel::RT v.hw () returns the homogenizing coordinate.

Note that you do not loose information with the homogeneous representation, because the FieldNumberType is a quotient.

 Kernel::FT v.x () returns the x-coordinate of v, that is hx/hw. Kernel::FT v.y () returns the y-coordinate of v, that is hy/hw.

The following operations are for convenience and for compatibility with higher dimensional vectors. Again they come in a Cartesian and homogeneous flavor.

 Kernel::RT v.homogeneous ( int i) returns the i'th homogeneous coordinate of v, starting with 0. Precondition: 0 i 2. Kernel::FT v.cartesian ( int i) returns the i'th Cartesian coordinate of v, starting at 0. Precondition: 0 i 1. Kernel::FT v.operator[] ( int i) returns cartesian(i). Precondition: 0 i 1. int v.dimension () returns the dimension (the constant 2). Direction_2 v.direction () returns the direction which passes through v. Vector_2 v.transform ( Aff_transformation_2 t) returns the vector obtained by applying t on v. Vector_2 v.perpendicular ( Orientation o) returns the vector perpendicular to v in clockwise or counterclockwise orientation.

### Operators

The following operations can be applied to vectors:

 Vector_2 v.operator+ ( w) Addition. Vector_2 v.operator- ( w) Subtraction. Vector_2 v.operator- () returns the opposite vector. Kernel::FT v.operator* ( w) returns the scalar product (= inner product) of the two vectors. Vector_2 operator* ( v, Kernel::RT s) Multiplication with a scalar from the right. Vector_2 operator* ( v, Kernel::FT s) Multiplication with a scalar from the right. Vector_2 operator* ( Kernel::RT s, v) Multiplication with a scalar from the left. Vector_2 operator* ( Kernel::FT s, v) Multiplication with a scalar from the left. Vector_2 v.operator/ ( Kernel::RT s) Division by a scalar.

Kernel::Vector_2