A Vector Type for the Plane

Code that works with points and directions in the plane is much easier to read when it can say 2 * (u + v) instead of scale(add(u, v), 2). Write a class Vec(x, y) that behaves like a mathematical vector:

• repr(v) is Vec(x, y) with the components written as Python would write them: Vec(1, 2), Vec(0.5, -3).
• u + v and u - v add and subtract componentwise; -v flips both signs.
• v * k, k * v and v / k scale by a number k.
• u == v is True when both components are equal; equal vectors must also count as one item in a set.
• abs(v) is the length sqrt(x**2 + y**2), and u.dot(v) is x1*x2 + y1*y2.
• A vector can be unpacked: x, y = v.
• Adding a vector and a number is an error: v + 3 raises TypeError.

Results are always new Vec objects; the operands never change. The setup has the helpers centroid(points), which computes sum(points, Vec(0, 0)) / len(points) with your operators, and raises(fn, *args), which returns the name of the exception a call raises (or None).

Examples

Input:  u, v = Vec(1, 2), Vec(3, -1)
        u + v, 2 * (u - v), abs(Vec(3, 4)), u.dot(v)
Output: (Vec(4, 1), Vec(-4, 6), 5.0, 1)

Input:  centroid([Vec(0, 0), Vec(4, 0), Vec(2, 6)]), len({Vec(1, 2), Vec(1, 2), Vec(2, 1)})
Output: (Vec(2.0, 2.0), 2)

Constraints

• Components are integers or floats; k is a number and k != 0 for division.
• The tests compare repr strings (so the outputs below are the repr of the results) and use ==, set, sorted(..., key=abs) and unpacking.

Goals

• Make `+`, `-`, `*`, `/`, `abs` and `==` work on your own objects
• Give a class a `__repr__` that looks like the code that builds the object
• Keep objects that define `==` usable in sets and as dictionary keys