# Vector definition

In maths vectors are considered a directed line segments(geometrically) and algebraically they are ordered n - tuples of real numbers. My questions are

1. How are these two definitions are equivalent?

2. What is the definition of vector addition in geometric definition?

3. What are the proofs of parallelogram law and triangle law of two vector addition? What is the difference between them?

• Analytic geometry links the geometrical definition with coordinates (i.e. numbers). – Mauro ALLEGRANZA Jan 21 at 17:02
• Actually, maths considers vectors to simply be elements of a structure called a vector space. Directed line segments and tuples of real numbers are only two of the many possible types of things that can be vectors. – amd Jan 21 at 21:22
• The tuples are in fact the dimensions of the directed line segments when it is drawn in space. – Faiq Irfan Jan 22 at 14:15
• @FaiqIrfan Tuples are different way to define vectors."Directed line segment" is a elementary definition so all the properties of vectors should be proved by this definition too, yes, indeed difficulty level may vary considerably. – prashant sharma Jan 23 at 1:47
• @amd I think vector space is a later development and vectors are elementary. So we must have some way answer to my questions only on the basis of vector elementary definition, that is, directed line segment. – prashant sharma Jan 23 at 1:51