## CGAL::Vector_3<Kernel>

### Definition

An object of the class Vector_3<Kernel> is a vector in the three-dimensional vector space  3. 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_3<Kernel> v ( Point_3<Kernel> a, Point_3<Kernel> b);
introduces the vector b-a.

Vector_3<Kernel> v ( Segment_3<Kernel> s);
introduces the vector s.target()-s.source().

Vector_3<Kernel> v ( Ray_3<Kernel> r);
introduces a vector having the same direction as r.

Vector_3<Kernel> v ( Line_3<Kernel> l);
introduces a vector having the same direction as l.

Vector_3<Kernel> v ( Null_vector NULL_VECTOR);
introduces a null vector v.

 Vector_3 v ( Kernel::RT hx, Kernel::RT hy, Kernel::FT hz, Kernel::RT hw = RT(1));
introduces a vector v initialized to (hx/hw, hy/hw, hz/hw).

### Operations

 bool v.operator== ( w) Test for equality: two vectors are equal, iff their x, y and z 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.hz () returns the homogeneous z 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. Kernel::FT v.z () returns the z coordinate of v, that is hz/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 3. Kernel::FT v.cartesian ( int i) returns the i'th Cartesian coordinate of v, starting at 0. Precondition: 0 i 2. Kernel::FT v.operator[] ( int i) returns cartesian(i). Precondition: 0 i 2. int v.dimension () returns the dimension (the constant 3). Vector_3 v.transform ( Aff_transformation_3 t) returns the vector obtained by applying t on v. Direction_3 v.direction () returns the direction of v.

### Operators

The following operations can be applied on vectors:

 Vector_3 v.operator+ ( w) Addition. Vector_3 v.operator- ( w) Subtraction. Vector_3 v.operator- () Negation. Kernel::FT v.operator* ( w) returns the scalar product (= inner product) of the two vectors. Vector_3 operator* ( v, Kernel::RT s) Multiplication with a scalar from the right. Vector_3 operator* ( v, Kernel::FT s) Multiplication with a scalar from the right. Vector_3 operator* ( Kernel::RT s, v) Multiplication with a scalar from the left. Vector_3 operator* ( Kernel::FT s, v) Multiplication with a scalar from the left. Vector_3 v.operator/ ( Kernel::RT s) Division by a scalar.