Originally Posted by Rollo87
I think what you're having the most trouble with is actually reading the question. What you need to see here is that r - r' is just one vector.
r is a vector from O to (4, 6, 8) and r' is another vector from O to (2, 8, 7). -r' will be the vector going from (2, 8, 7) to O. If this gets translated so that the starting end is at the origin, it will go from O to (-2, -8, -7). So r - r' is the sum of the vectors r and -r' and represents the vector going from (4, 6, 8) to (-2, -8, -7), which can be found by adding the ordinates, giving (2, -2, 1).
a) Now that we have the vector r - r' = (2, -2, 1), we can work out it's magnitude (length) by summing the squares of the i, j, k components and then taking the square root.
i.e. |r - r'| = sqrt[2^2 + (-2)^2 + 1^2]
= sqrt(4 + 4 + 1)
We can disregard the negative answer because when dealing with vectors, the modulus is the LENGTH of the vector, and we can't have a negative length.
Now we can find (r - r')hat by dividing each component of r - r' by the length of the vector. What we are really doing here is finding a vector in the direction of r - r', but with a length of 1 unit.
So (r - r')hat = (2/3, -2/3, 1/3).
b) doesn't make any sense because 2(r - r') is a vector and the divergence of (r-r') is a scalar, and like you say, r-r' is not a function of x, y and z... Are you sure you copied down the question correctly?