Verify using an example that a + ( b• c) = (a •b ) +c ? Explain your reasoning. I have been over this a thousand times and I don't understand (the notes I was given online are incorrect so I don't know how to answer this). THANKS!
Ok, well both sides are equal yes, but you can't actually do what they are asking right? Because you do the brackets first, and a dot product provides a scalar answer, therefore, you can't add a scalar to a vector, because that is meaningless. Is this correct?
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)
To verify the equation a + (b • c) = (a • b) + c, we can consider a specific example. Let's assign some values to the variables:
Let a = 2, b = 3, and c = 4.
Now let's substitute these values into the equation on both sides:
Left-hand side (LHS): a + (b • c)
= 2 + (3 • 4)
= 2 + 12
= 14
Right-hand side (RHS): (a • b) + c
= (2 • 3) + 4
= 6 + 4
= 10
By evaluating both sides of the equation with the given values, we see that the LHS is equal to 14 and the RHS is equal to 10. Therefore, the equation a + (b • c) = (a • b) + c does not hold in this example.
This means that the equation is not universally true for all values of a, b, and c. There might be specific cases where it holds, but it doesn't hold for the example we considered.
It's important to note that this example doesn't prove the equation is false in general since we only looked at one specific set of values for a, b, and c. To fully verify the equation's validity, you would need to show that it holds for all possible values of a, b, and c.