Originally Posted by
Rollo87 I'll give you more detail (should've just done this in the first place). This is the actual quesiotn:
1. Consider the separation vector
r - r ' from the point r' = (x1 ', x2 ', x3 ')= (2, 8, 7) to the point r
= (x1, x2, x3)= (4, 6, 8)expressed in Cartesian coordinates. (a) Determine the magnitude (r - r' ), and construct the unit vector (r - r ')(hat)
= r - r' /|r - r' |.
(b) Show that “DEL(r - r ')^2= 2 (r - r').
Obviouly constructing the unit vector is very simple. And yes i know how to take both the dot product and cross-product but my problem is more that I'm not sure what the question wants me to do exactly. Also it's the dot product of Del with (r-r') so it's the divergence I'm looking for and I know how to do this but all the stuff i've read gives it in terms of partial derivatives which is fine but these vectors are integers they are not functions of x,y or z.
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)
= sqrt(9)
= 3
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?