if r= <x,y,z> and r_0= <x_0,y_0,z_0>, describe the set of all points (x,y,z) such that magnitude[r-r_0]=4

r-r0 is just a vector from r0 to r. So, what you have is a sphere of radius 4 centered at (x0,y0,z0)

To describe the set of all points (x, y, z) such that magnitude[r - r_0] = 4, we need to find the distance between vector r and vector r_0 and set it equal to 4.

The magnitude of a vector is given by taking the square root of the sum of the squares of its components. In this case, the magnitude of (r - r_0) is equal to 4.

So, we can write the equation as:

|<x-x_0, y-y_0, z-z_0>| = 4

To solve this equation, we can square both sides:

|<x-x_0, y-y_0, z-z_0>|^2 = 4^2

Using the definition of magnitude, we have:

(x - x_0)^2 + (y - y_0)^2 + (z - z_0)^2 = 16

Now, we can expand and simplify the equation:

x^2 - 2x*x_0 + x_0^2 + y^2 - 2y*y_0 + y_0^2 + z^2 - 2z*z_0 + z_0^2 = 16

Rearranging terms and combining like terms, we get:

x^2 + y^2 + z^2 - 2x*x_0 - 2y*y_0 - 2z*z_0 + x_0^2 + y_0^2 + z_0^2 - 16 = 0

This equation represents a sphere in 3D space, centered at the point (x_0, y_0, z_0) with a radius of 4 units. Thus, the set of all points (x, y, z) that satisfy the equation represents a sphere of radius 4 centered at the point (x_0, y_0, z_0).