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I doing programming but I am not good at math.

I want to represent point and vector in different type. Both point and vector are tuple of 3 elements, where point represent location and vector represent direction and magnitude.

Given P for point, V for vector and S for one dimensional scalar type, how they interact each other in arithmetic operations?

By arithmetic operations I mean addition, subtraction, multiplication dan division.

In my understanding, adding or subtract 2 vectors will get result as vector.

V + V = V
V - V = V

but, what about addition or subtraction vector with scalar? is that make sense?

Multiplication vector with scalar have meaning to scale the magnitude of vector, which is give vector as result.

V * S = V

Subtraction of 2 points will result a vector between two points

P - P = V

well, that is what I know, I not sure about the other operators.

So, my question is, what is the result type of arithmetic operations of vectors, points and scalars?

V + V = V      V + S = ?      V + P = ?
V - V = V      V - S = ?      V - P = ?
V * V = ?      V * S = V      V * P = ?
V / V = ?      V / S = V      V / P = ?

P + V = ?      P + S = ?      P + P = ?
P - V = ?      P - S = ?      P - P = V
P * V = ?      P * S = ?      P * P = ?
P / V = ?      P / S = ?      P / P = ?

S + V = ?      S + S = ?      S + P = ?
S - V = ?      S - S = ?      S - P = ?
S * V = V      S * S = ?      S * P = ?
S / V = V      S / S = ?      S / P = ?

P: point

V: vector

S: scalar

X: meaningless/not make sense

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