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Ok here goes:

*only considering vectors in two-dimensions*

We define a vector to be a directed line from A to B.

We define equality between two vectors if we can use translations to bring the two starting points and two final points to coincide.

Remark 1)

Any vector can be defined by two parameters, its tensor and its unit vector.

The unit vector has length 1 and direction.

Direction can be specified by the angle which it makes with the x-axis, restricted to 0 and 2pi and measured anticlockwise from the x-axis

Thus, to identify a vector, all we need is an tensor and an angle.

Remark 2)

We take two vectors, a and b. We wish to look at what is meant by the quotient of a and b.

q = a/b, that is q * b = a.

q being a quantity such that when the vector b is left multiplied by q, it becomes a.

Geometrically, q changes b by 1) scaling its length 2) changing its direction.

The scaling length is what is meant by a tensor.

The change in direction is precisely a angle which is turned through, anti clockwise.

Claim:

q is a vector as it has precisely tensor and angle.

What I am stuck on now is explicitly relating this to complex numbers.

A complex number can be expressed as re^ix where r is a positive real number and x is a number between 0 and 2pi. Thus, it is much like length and angle.

I can see that there is a bijection between complex numbers and vectors.

I can also see that with the bijection, addition and subtraction hold. So there is a isomorphism between vectors and complex numbers.

But how on earth do I begin to relate the multiplication of complex numbers to vectors?

Is it as simple as saying, define multiplication like that which we get some complex number multiplication. But how do I go about showing that this is a good definition.... look at distributivity, etc? (not communitivity?)

Thanks in advance

Joe V