# class Vector

The `Vector` class represents a mathematical vector, which is useful in its own right, and also constitutes a row or column of a Matrix.

## Method Catalogue¶ ↑

To create a Vector:

To access elements:

To enumerate the elements:

Properties of vectors:

Vector arithmetic:

• #*(x) “is matrix or number”

• #+(v)

• #-(v)

• #+@

• #-@

Vector functions:

Conversion to other data types:

String representations:

### Attributes

elements[R]

INSTANCE CREATION

### Public Class Methods

[](*array) click to toggle source

Creates a Vector from a list of elements.

`Vector[7, 4, ...]`
```# File lib/matrix.rb, line 1741
def Vector.[](*array)
new convert_to_array(array, false)
end```
basis(size:, index:) click to toggle source

Returns a standard basis `n`-vector, where k is the index.

`Vector.basis(size:, index:) # => Vector[0, 1, 0]`
```# File lib/matrix.rb, line 1758
def Vector.basis(size:, index:)
raise ArgumentError, "invalid size (#{size} for 1..)" if size < 1
raise ArgumentError, "invalid index (#{index} for 0...#{size})" unless 0 <= index && index < size
array = Array.new(size, 0)
array[index] = 1
new convert_to_array(array, false)
end```
elements(array, copy = true) click to toggle source

Creates a vector from an Array. The optional second argument specifies whether the array itself or a copy is used internally.

```# File lib/matrix.rb, line 1749
def Vector.elements(array, copy = true)
new convert_to_array(array, copy)
end```
independent?(*vs) click to toggle source

Returns `true` iff all of vectors are linearly independent.

```Vector.independent?(Vector[1,0], Vector[0,1])
=> true

Vector.independent?(Vector[1,2], Vector[2,4])
=> false```
```# File lib/matrix.rb, line 1851
def Vector.independent?(*vs)
vs.each do |v|
raise TypeError, "expected Vector, got #{v.class}" unless v.is_a?(Vector)
Vector.Raise ErrDimensionMismatch unless v.size == vs.first.size
end
return false if vs.count > vs.first.size
Matrix[*vs].rank.eql?(vs.count)
end```
new(array) click to toggle source

::new is private; use Vector[] or ::elements to create.

```# File lib/matrix.rb, line 1769
def initialize(array)
# No checking is done at this point.
@elements = array
end```

### Public Instance Methods

*(x) click to toggle source

Multiplies the vector by `x`, where `x` is a number or another vector.

```# File lib/matrix.rb, line 1911
def *(x)
case x
when Numeric
els = @elements.collect{|e| e * x}
self.class.elements(els, false)
when Matrix
Matrix.column_vector(self) * x
when Vector
Vector.Raise ErrOperationNotDefined, "*", self.class, x.class
else
apply_through_coercion(x, __method__)
end
end```
+(v) click to toggle source

```# File lib/matrix.rb, line 1928
def +(v)
case v
when Vector
Vector.Raise ErrDimensionMismatch if size != v.size
els = collect2(v) {|v1, v2|
v1 + v2
}
self.class.elements(els, false)
when Matrix
Matrix.column_vector(self) + v
else
apply_through_coercion(v, __method__)
end
end```
+@() click to toggle source
```# File lib/matrix.rb, line 1976
def +@
self
end```
-(v) click to toggle source

Vector subtraction.

```# File lib/matrix.rb, line 1946
def -(v)
case v
when Vector
Vector.Raise ErrDimensionMismatch if size != v.size
els = collect2(v) {|v1, v2|
v1 - v2
}
self.class.elements(els, false)
when Matrix
Matrix.column_vector(self) - v
else
apply_through_coercion(v, __method__)
end
end```
-@() click to toggle source
```# File lib/matrix.rb, line 1980
def -@
collect {|e| -e }
end```
/(x) click to toggle source

Vector division.

```# File lib/matrix.rb, line 1964
def /(x)
case x
when Numeric
els = @elements.collect{|e| e / x}
self.class.elements(els, false)
when Matrix, Vector
Vector.Raise ErrOperationNotDefined, "/", self.class, x.class
else
apply_through_coercion(x, __method__)
end
end```
==(other) click to toggle source

Returns `true` iff the two vectors have the same elements in the same order.

```# File lib/matrix.rb, line 1880
def ==(other)
return false unless Vector === other
@elements == other.elements
end```
[](i) click to toggle source

Returns element number `i` (starting at zero) of the vector.

```# File lib/matrix.rb, line 1779
def [](i)
@elements[i]
end```
Also aliased as: element, component
angle_with(v) click to toggle source

Returns an angle with another vector. Result is within the [0…Math::PI].

```Vector[1,0].angle_with(Vector[0,1])
# => Math::PI / 2
```
```# File lib/matrix.rb, line 2085
def angle_with(v)
raise TypeError, "Expected a Vector, got a #{v.class}" unless v.is_a?(Vector)
Vector.Raise ErrDimensionMismatch if size != v.size
prod = magnitude * v.magnitude
raise ZeroVectorError, "Can't get angle of zero vector" if prod == 0

Math.acos( inner_product(v) / prod )
end```
clone() click to toggle source

Returns a copy of the vector.

```# File lib/matrix.rb, line 1893
def clone
self.class.elements(@elements)
end```
coerce(other) click to toggle source

The coerce method provides support for Ruby type coercion. This coercion mechanism is used by Ruby to handle mixed-type numeric operations: it is intended to find a compatible common type between the two operands of the operator. See also Numeric#coerce.

```# File lib/matrix.rb, line 2134
def coerce(other)
case other
when Numeric
return Matrix::Scalar.new(other), self
else
raise TypeError, "#{self.class} can't be coerced into #{other.class}"
end
end```
collect() { |e| ... } click to toggle source

Like Array#collect.

```# File lib/matrix.rb, line 2040
def collect(&block) # :yield: e
els = @elements.collect(&block)
self.class.elements(els, false)
end```
Also aliased as: map
collect2(v) { |e1, e2| ... } click to toggle source

Collects (as in Enumerable#collect) over the elements of this vector and `v` in conjunction.

```# File lib/matrix.rb, line 1829
def collect2(v) # :yield: e1, e2
raise TypeError, "Integer is not like Vector" if v.kind_of?(Integer)
Vector.Raise ErrDimensionMismatch if size != v.size
Array.new(size) do |i|
yield @elements[i], v[i]
end
end```
component(i)
Alias for: []
covector() click to toggle source

Creates a single-row matrix from this vector.

```# File lib/matrix.rb, line 2101
def covector
Matrix.row_vector(self)
end```
cross(*vs)
Alias for: cross_product
cross_product(*vs) click to toggle source

Returns the cross product of this vector with the others.

```Vector[1, 0, 0].cross_product Vector[0, 1, 0]   => Vector[0, 0, 1]
```

It is generalized to other dimensions to return a vector perpendicular to the arguments.

```Vector[1, 2].cross_product # => Vector[-2, 1]
Vector[1, 0, 0, 0].cross_product(
Vector[0, 1, 0, 0],
Vector[0, 0, 1, 0]
)  #=> Vector[0, 0, 0, 1]
```
```# File lib/matrix.rb, line 2015
def cross_product(*vs)
raise ErrOperationNotDefined, "cross product is not defined on vectors of dimension #{size}" unless size >= 2
raise ArgumentError, "wrong number of arguments (#{vs.size} for #{size - 2})" unless vs.size == size - 2
vs.each do |v|
raise TypeError, "expected Vector, got #{v.class}" unless v.is_a? Vector
Vector.Raise ErrDimensionMismatch unless v.size == size
end
case size
when 2
Vector[-@elements[1], @elements[0]]
when 3
v = vs[0]
Vector[ v[2]*@elements[1] - v[1]*@elements[2],
v[0]*@elements[2] - v[2]*@elements[0],
v[1]*@elements[0] - v[0]*@elements[1] ]
else
rows = self, *vs, Array.new(size) {|i| Vector.basis(size: size, index: i) }
Matrix.rows(rows).laplace_expansion(row: size - 1)
end
end```
Also aliased as: cross
dot(v)
Alias for: inner_product
each(&block) click to toggle source

Iterate over the elements of this vector

```# File lib/matrix.rb, line 1806
def each(&block)
@elements.each(&block)
self
end```
each2(v) { |e1, e2| ... } click to toggle source

Iterate over the elements of this vector and `v` in conjunction.

```# File lib/matrix.rb, line 1815
def each2(v) # :yield: e1, e2
raise TypeError, "Integer is not like Vector" if v.kind_of?(Integer)
Vector.Raise ErrDimensionMismatch if size != v.size
size.times do |i|
yield @elements[i], v[i]
end
self
end```
element(i)
Alias for: []
elements_to_f() click to toggle source
```# File lib/matrix.rb, line 2112
def elements_to_f
warn "#{caller(1)[0]}: warning: Vector#elements_to_f is deprecated"
map(&:to_f)
end```
elements_to_i() click to toggle source
```# File lib/matrix.rb, line 2117
def elements_to_i
warn "#{caller(1)[0]}: warning: Vector#elements_to_i is deprecated"
map(&:to_i)
end```
elements_to_r() click to toggle source
```# File lib/matrix.rb, line 2122
def elements_to_r
warn "#{caller(1)[0]}: warning: Vector#elements_to_r is deprecated"
map(&:to_r)
end```
eql?(other) click to toggle source
```# File lib/matrix.rb, line 1885
def eql?(other)
return false unless Vector === other
@elements.eql? other.elements
end```
hash() click to toggle source

Returns a hash-code for the vector.

```# File lib/matrix.rb, line 1900
def hash
@elements.hash
end```
independent?(*vs) click to toggle source

Returns `true` iff all of vectors are linearly independent.

```Vector[1,0].independent?(Vector[0,1])
=> true

Vector[1,2].independent?(Vector[2,4])
=> false```
```# File lib/matrix.rb, line 1869
def independent?(*vs)
self.class.independent?(self, *vs)
end```
inner_product(v) click to toggle source

Returns the inner product of this vector with the other.

```Vector[4,7].inner_product Vector[10,1]  => 47
```
```# File lib/matrix.rb, line 1992
def inner_product(v)
Vector.Raise ErrDimensionMismatch if size != v.size

p = 0
each2(v) {|v1, v2|
p += v1 * v2.conj
}
p
end```
Also aliased as: dot
inspect() click to toggle source

Overrides Object#inspect

```# File lib/matrix.rb, line 2157
def inspect
"Vector" + @elements.inspect
end```
magnitude() click to toggle source

Returns the modulus (Pythagorean distance) of the vector.

`Vector[5,8,2].r => 9.643650761`
```# File lib/matrix.rb, line 2051
def magnitude
Math.sqrt(@elements.inject(0) {|v, e| v + e.abs2})
end```
Also aliased as: r, norm
map()
Alias for: collect
map2(v) { |e1, e2| ... } click to toggle source

Like #collect2, but returns a Vector instead of an Array.

```# File lib/matrix.rb, line 2060
def map2(v, &block) # :yield: e1, e2
els = collect2(v, &block)
self.class.elements(els, false)
end```
norm()
Alias for: magnitude
normalize() click to toggle source

Returns a new vector with the same direction but with norm 1.

```v = Vector[5,8,2].normalize
# => Vector[0.5184758473652127, 0.8295613557843402, 0.20739033894608505]
v.norm => 1.0```
```# File lib/matrix.rb, line 2074
def normalize
n = magnitude
raise ZeroVectorError, "Zero vectors can not be normalized" if n == 0
self / n
end```
r()
Alias for: magnitude
size() click to toggle source

Returns the number of elements in the vector.

```# File lib/matrix.rb, line 1795
def size
@elements.size
end```
to_a() click to toggle source

Returns the elements of the vector in an array.

```# File lib/matrix.rb, line 2108
def to_a
@elements.dup
end```
to_s() click to toggle source

Overrides Object#to_s

```# File lib/matrix.rb, line 2150
def to_s
"Vector[" + @elements.join(", ") + "]"
end```

### Private Instance Methods

[]=(i, v) click to toggle source
```# File lib/matrix.rb, line 1785
def []=(i, v)
@elements[i]= v
end```
Also aliased as: set_element, set_component
set_component(i, v)
Alias for: []=
set_element(i, v)
Alias for: []=