## 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<Kernel> v ( Point_2<Kernel> a, Point_2<Kernel> b);
| introduces the vector b-a.
Vector_2<Kernel> v ( Segment_2<Kernel> s);
| introduces the vector s.target()-s.source().
Vector_2<Kernel> v ( Ray_2<Kernel> r);
| introduces the vector having the same direction as r.
Vector_2<Kernel> v ( Line_2<Kernel> l);
| introduces the vector having the same direction as l.
Vector_2<Kernel> v ( Null_vector NULL_VECTOR);
| introduces a null vector v.
Vector_2<Kernel> 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<Kernel>
| v.direction () | returns the direction which passes through v.
Vector_2<Kernel> | v.transform ( Aff_transformation_2<Kernel> t)
|  | returns the vector obtained by applying t on v.
Vector_2<Kernel> | 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<Kernel> | v.operator+ ( w) | Addition.
Vector_2<Kernel> | v.operator- ( w) | Subtraction.
Vector_2<Kernel> | v.operator- () | returns the opposite vector.
Kernel::FT | v.operator* ( w) | returns the scalar product (= inner product) of the two vectors.
Vector_2<Kernel> | operator* ( v, Kernel::RT s)
|  | Multiplication with a scalar from the right.
Vector_2<Kernel> | operator* ( v, Kernel::FT s)
|  | Multiplication with a scalar from the right.
Vector_2<Kernel> | operator* ( Kernel::RT s, v)
|  | Multiplication with a scalar from the left.
Vector_2<Kernel> | operator* ( Kernel::FT s, v)
|  | Multiplication with a scalar from the left.
Vector_2<Kernel> | v.operator/ ( Kernel::RT s)
|  | Division by a scalar.

### See Also

Kernel::Vector_2