Copyright | (c) Antony Courtney and Henrik Nilsson Yale University 2003 |
---|---|

License | BSD-style (see the LICENSE file in the distribution) |

Maintainer | ivan.perez@keera.co.uk |

Stability | provisional |

Portability | non-portable (GHC extensions) |

Safe Haskell | Safe |

Language | Haskell98 |

Vector space type relation and basic instances.

## Synopsis

- class (Eq a, Floating a) => VectorSpace v a | v -> a where
- zeroVector :: v
- (*^) :: a -> v -> v
- (^/) :: v -> a -> v
- (^+^) :: v -> v -> v
- (^-^) :: v -> v -> v
- negateVector :: v -> v
- dot :: v -> v -> a
- norm :: v -> a
- normalize :: v -> v

# Documentation

class (Eq a, Floating a) => VectorSpace v a | v -> a where #

Vector space type relation.

A vector space is a set (type) closed under addition and multiplication by
a scalar. The type of the scalar is the *field* of the vector space, and
it is said that `v`

is a vector space over `a`

.

The encoding uses a type class |VectorSpace| `v a`

, where `v`

represents
the type of the vectors and `a`

represents the types of the scalars.

zeroVector, (*^), (^+^), dot

zeroVector :: v #

Vector with no magnitude (unit for addition).

(*^) :: a -> v -> v infixr 9 #

Multiplication by a scalar.

(^/) :: v -> a -> v infixl 9 #

Division by a scalar.

(^+^) :: v -> v -> v infixl 6 #

Vector addition

(^-^) :: v -> v -> v infixl 6 #

Vector subtraction

negateVector :: v -> v #

Vector negation. Addition with a negated vector should be same as subtraction.

Dot product (also known as scalar or inner product).

For two vectors, mathematically represented as `a = a1,a2,...,an`

and ```
b
= b1,b2,...,bn
```

, the dot product is ```
a . b = a1*b1 + a2*b2 + ... +
an*bn
```

.

Some properties are derived from this. The dot product of a vector with
itself is the square of its magnitude (`norm`

), and the dot product of
two orthogonal vectors is zero.

Vector's norm (also known as magnitude).

For a vector represented mathematically as `a = a1,a2,...,an`

, the norm
is the square root of `a1^2 + a2^2 + ... + an^2`

.

Return a vector with the same origin and orientation (angle), but such that the norm is one (the unit for multiplication by a scalar).

## Instances

VectorSpace Double Double # | |

Defined in Data.VectorSpace | |

VectorSpace Float Float # | |

Defined in Data.VectorSpace | |

RealFloat a => VectorSpace (Vector3 a) a # | |

RealFloat a => VectorSpace (Vector2 a) a # | |

(Eq a, Floating a) => VectorSpace (a, a) a # | Vector space instance for pairs of |

Defined in Data.VectorSpace | |

(Eq a, Floating a) => VectorSpace (a, a, a) a # | Vector space instance for triplets of |

Defined in Data.VectorSpace | |

(Eq a, Floating a) => VectorSpace (a, a, a, a) a # | Vector space instance for tuples with four |

Defined in Data.VectorSpace zeroVector :: (a, a, a, a) # (*^) :: a -> (a, a, a, a) -> (a, a, a, a) # (^/) :: (a, a, a, a) -> a -> (a, a, a, a) # (^+^) :: (a, a, a, a) -> (a, a, a, a) -> (a, a, a, a) # (^-^) :: (a, a, a, a) -> (a, a, a, a) -> (a, a, a, a) # negateVector :: (a, a, a, a) -> (a, a, a, a) # | |

(Eq a, Floating a) => VectorSpace (a, a, a, a, a) a # | Vector space instance for tuples with five |

Defined in Data.VectorSpace zeroVector :: (a, a, a, a, a) # (*^) :: a -> (a, a, a, a, a) -> (a, a, a, a, a) # (^/) :: (a, a, a, a, a) -> a -> (a, a, a, a, a) # (^+^) :: (a, a, a, a, a) -> (a, a, a, a, a) -> (a, a, a, a, a) # (^-^) :: (a, a, a, a, a) -> (a, a, a, a, a) -> (a, a, a, a, a) # negateVector :: (a, a, a, a, a) -> (a, a, a, a, a) # dot :: (a, a, a, a, a) -> (a, a, a, a, a) -> a # |