# How can I prove that the dot product is distributive?

• Mathematicsresear
In summary, the task is to prove the properties related to the dot product of vectors A and B, and the relationship between A.(B+C) and A.B+A.C. The homework equations state that the dot product can be defined as the product of the lengths of the vectors and the cosine of the angle between them. The task also requires a generalization for an arbitrary coordinate system, without assuming the standard basis vectors. It is suggested to first prove the properties for cartesian coordinates and then extend it to other coordinate systems through transformations.f

## 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.

## 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?