How can I prove that the dot product is distributive?

[Mathematicsresear]

Homework Statement

Prove that for vectors A,B and C, A.(B+C)=A.B+A.C and prove the property that for two vectors, A and B the dot product is equal to A^ie_i . B^je_j = e_i.e_jA^iB^j

Homework Equations

Only use the definition where for two vectors a and b the (length of a)(length of b)cost =a.b
Generalize for an arbitrary coordinate system (not necessarily cartesian). Moreover, I'm not allowed to assume that the basis are the standard, I,j,k basis vectors. So the metric tensor is not the kronecker delta. I mean that I shouldn't assume that I'm just working in cartesian coordinates.

The Attempt at a Solution

For the first, I got length of A length ofB+C cost but I'm not sure what to do after that.

 

[Ray Vickson]

Homework Statement

Prove that for vectors A,B and C, A.(B+C)=A.B+A.C and prove the property that for two vectors, A and B the dot product is equal to A^ie_i . B^je_j = e_i.e_jA^iB^j

Homework Equations

Only use the definition where for two vectors a and b the (length of a)(length of b)cost =a.b
Generalize for an arbitrary coordinate system (not necessarily cartesian). Moreover, I'm not allowed to assume that the basis are the standard, I,j,k basis vectors. So the metric tensor is not the kronecker delta. I mean that I shouldn't assume that I'm just working in cartesian coordinates.

The Attempt at a Solution

For the first, I got length of A length ofB+C cost but I'm not sure what to do after that.

Why not prove it first for cartesian coordinates, then develop the expression in other coordinate systems by transformations?