How can this be proven! I have tried so many ways! PLEASE help! Verify using an example that Vector a + (Vector b • Vector c) = (Vector a • Vector b) + Vector c? Explain your reasoning.

Answer 1

There is something wrong with what you are trying to prove.

"Vector a" is a vector, but (Vector b • Vector c) is a scalar (just a number). I am assuming you really meant to write a dot product when you wrote Vector b • Vector c.

You cannot add a scalar to a vector.

Answer 2

THis is what I was told to do, I am pretty sure that there are typos in the assignment I was given. But thanks anyways.

Answer 3

I think you mean the identity:

a dot ( b cross c) =

c dot (a cross b)

This can most easily be proved byintroducing the anti-symmetric pseudo tensor e_{i,j,k}:

e_{i,j,k} = 1 if (i,j,k) is a cyclic permutation of (1,2,3)

e_{i,j,k} = -1 if (i,j,k) is a cyclic permutation of (3,2,1)

In all other cases e_{i,j,k} = 0

e_{i,j,k} is then zero if two or more of its indices are equal to each other. If you interchange two indices then it changes sign.

The cross product can be writen as follows:

if Z = X cross Y, then:

Z_i = e_{i,j,k} X_j Y_k

Where we sum over the repeated indices j and k, his is called "Einstein summation convention".

The dot product between vectors X and Y can be written as:

X_i Y_i

where again the Einstein summaton convention is used and we sum over the repeated index i.

Let's now write out the term:

a dot ( b cross c)

We can write this as:

a_i (b cross c)_i.

Inserting

(b cross c)_i = e_{i,j,k} b_j c_k gives:

a dot ( b cross c) =

e_{i,j,k} a_i b_j c_k

If you now use that the e_{i,j,k} tensor is cyclically symmetric:

a dot ( b cross c) =

e_{i,j,k} a_i b_j c_k =

e_{k,i,j}c_k a_i b_j =

c dot (a cross b)