Vectors

A vector is an ordered list of numbers. In panchi, Vector is the fundamental object representing a point or direction in n-dimensional space.

Construction

import panchi as pan

v = pan.Vector([1, 2, 3])
print(v)       # [1, 2, 3]
print(v.dims)  # 3

Vectors hold numbers (int, float, or Fraction). You can write fractions as strings:

v = pan.Vector(["1/3", "2/3", "1/2"])
print(v)  # [1/3, 2/3, 1/2]

For full details on exact arithmetic, see Exact Arithmetic.

In two dimensions, a Vector is an arrow from the origin. panchi can draw them for you:

Three vectors plotted with Animator2D.plot_vectors. See Visualizations.

Arithmetic

Vectors support standard arithmetic with natural syntax:

a = pan.Vector([1, 2, 3])
b = pan.Vector([4, 5, 6])

print(a + b)   # [5, 7, 9]
print(a - b)   # [-3, -3, -3]
print(2 * a)   # [2, 4, 6]
print(a * 2)   # [2, 4, 6]  — same as above
print(a / 2)   # [0.5, 1.0, 1.5]
print(-a)      # [-1, -2, -3]

All operations return a new Vector and leave the original unchanged.

Addition places the two arrows tip-to-tail; scalar multiplication stretches an arrow along its own direction:

animate_addition(Vector([3, 1]), Vector([1, 3]))

animate_scaling(Vector([2, 1]), scale_factor=2.0)

Magnitude and normalization

The magnitude of a vector is its Euclidean length — the square root of the sum of squared components.

v = pan.Vector([3, 4])
print(v.magnitude)   # 5.0
print(v.normalize()) # [0.6, 0.8]

normalize() returns the unit vector pointing in the same direction. It raises ZeroDivisionError if the vector has zero magnitude.

Dot product

The dot product of two vectors measures how much they point in the same direction. It is defined as the sum of the products of corresponding components.

\[\mathbf{u} \cdot \mathbf{v} = \sum_{i} u_i v_i\]

u = pan.Vector([1, 2, 3])
v = pan.Vector([4, 5, 6])
print(pan.dot(u, v))  # 32

If the dot product is zero, the vectors are orthogonal.

Cross product

The cross product is defined only for 3D vectors. It produces a new vector perpendicular to both inputs, with magnitude equal to the area of the parallelogram they span.

u = pan.Vector([1, 0, 0])
v = pan.Vector([0, 1, 0])
print(pan.cross(u, v))  # [0, 0, 1]

Factory functions

panchi provides several convenience functions for constructing common vectors:

pan.zero_vector(3)      # [0, 0, 0]
pan.one_vector(3)       # [1, 1, 1]
pan.unit_vector(3, 1)   # [0, 1, 0]  — standard basis vector e₁
pan.random_vector(3)    # random entries

Indexing and iteration

Vectors are indexable and iterable:

v = pan.Vector([10, 20, 30])
print(v[0])            # 10
v[1] = 99              # mutation is supported
print(list(v))         # [10, 99, 30]

Conversion

v = pan.Vector([1, 2, 3])
v.to_list()    # [1, 2, 3]  — returns an independent copy
v.to_tuple()   # (1, 2, 3)