Vectors

General vector class

Name

Vector

Inheritance

array of doubles: Array<double>

Constituents

• Number of elements, n.
• Dynamically allocated array, indexed from base 0 using square brackets.

Methods

Creation

• Vector(number_of_elements)

  e.g. Vector x(3); or y = Vector(n);.

• Vector(existing_vector_to_copy)

  e.g. Vector y; cin >> y; Vector x(y);

  This should be faster than initialising a new vector and then copying another.

• Create(number_of_elements)

  e.g. Vector x; x.Create(7);

Input and output

(Array class inherits from Classio and provides Read and Write operations which allow the << and >> operators to be used, in which case the number of elements is read from and written to the I/O stream. Array also provides Read_elements and Write_elements methods to read and write the element list only, without the number of elements.) e.g.

Vector x; cin >> x; cout << x << endl; x.Write_elements(cout);

Assignment

=

e.g. Vector x,y; cin >> x; y = x;

Extract a sub-vector

   Vector subvector(
      const int i0, // first index of subvector
      const int i1  // last index of subvector
   ) const

e.g. Vector y = x.subvector(0,4);

Inner product

The following methods all return a.b, where a and b are vectors with the same number of elements.

• friend double inner_product(const Vector& a, const Vector& b)
• friend double dot(const Vector& a, const Vector& b)
• friend double operator*(const Vector& a, const Vector& b)

e.g.

   Vector x,y; cin >> x >> y;
   double a = inner_product(x,y), b = dot(x,y), c = x * y; // a = b = c

Norm

(Sum of squares of elements.) friend double norm(const Vector& a)

e.g. vector a; cin >> a; double anorm = norm(a);

Length

friend double length(const Vector& a)

e.g. Vector a; cin >> a; double alen = length(a);

Normalise

(Scale to unit length.) friend void normalise(Vector& a)

e.g. vector a; cin >> a; normalise(a); cout << a;

Arithmetic operations

These methods allow conventional-looking equations to be written with vector quantities.

• Add (a + b): friend Vector operator+(const Vector& a, const Vector& b)
• Subtract (a - b): friend Vector operator-(const Vector& a, const Vector& b)
• Negate (-a): friend Vector operator-(const Vector& a)
• Multiply by a scalar, integer or real (s * a or a * s):
  • friend Vector operator*(const Vector& a, const int s)
  • friend Vector operator*(const int s, const Vector& a)
  • friend Vector operator*(const Vector& a, const double s)
  • friend Vector operator*(const double s, const Vector& a)
• Divide by a scalar, integer or real (a / s):
  • friend Vector operator/(const Vector& a, const int s)
  • friend Vector operator/(const Vector& a, const double s)

General vector class (fixed dimension)

Name

Vector_fixed_dimension

Inheritance

vector

This class is the base for vectors of a specific dimension.

4D vector class

Name

Vector4d

Inheritance

Vector_fixed_dimension

3D vector class

Name

Vector3d

Inheritance

Vector_fixed_dimension

Methods

Cross product

Calculate the vector (cross) product of two vectors.

• friend Vector3d cross_product(const Vector3d& a, const Vector3d& b)
• friend Vector3d operator^(const Vector3d& a, const Vector3d& b)

Vector triple product

Calculates the volume a.(b^c).

   friend double vector_triple_product(const Vector3d& a, const Vector3d& b,
      const Vector3d& c)

Rotation

Axis vectors must be of unit length.

• About an arbitrary axis (e.g. for rotating velocity vectors)

     friend Vector3d rotate(
        const Vector3d& a,    // initial vector
        const Vector3d& axis, // axis vector (rotation is right-hand screw)
        const double phi      // rotation angle in radians
     )
  

• About an arbitrary axis through an arbitrary point (e.g. for rotating position vectors)

     friend Vector3d rotate_about(
        const Vector3d& a,    // initial vector
        const Vector3d& axis, // axis vector (rotation is right-hand screw)
        const Vector3d& p,    // point through which axis passes
        const double phi      // rotation angle in radians
     )
  

2D vector class

Name

Vector2d

Inheritance

Vector_fixed_dimension

Methods

Cross product

Calculates the real value a^b.

   friend double operator^(const Vector2d& a, const Vector2d& b)

Rotation

•    friend Vector2d rotate(
        const Vector2d& a,    // initial vector
        const double phi      // rotation angle in radians (anticlockwise)
     )
  

• About an arbitrary axis through an arbitrary point (e.g. for rotating position vectors)

     friend Vector2d rotate_about(
        const Vector2d& a,    // initial vector
        const Vector2d& p,    // point through which axis passes
        const double phi      // rotation angle in radians (anticlockwise)
     )
  

General real vector class

Name

Real_vector

Inheritance

Arithmetic_matrix<double>