Question

Let \(\mathbf{r}=\langle x, y, z\rangle\) and \(\mathbf{r}_{0}=\langle 1,1,1\rangle .\) Describe the set of all points \((x, y, z)\) such that \(\left\|\mathbf{r}-\mathbf{r}_{0}\right\|=2\)

Short answer

The set of all points \((x, y, z)\) such that \(\|\mathbf{r}-\mathbf{r}_{0}\|=2\) is described by the equation \((x-1)^2 + (y-1)^2 + (z-1)^2 = 4\), which is a sphere of radius 2 centered at the point \((1,1,1)\).

Step-by-step solution

Find the vector difference

Find the vector \(\mathbf{r}-\mathbf{r}_{0}\) by subtracting the coordinates of \(\mathbf{r}_{0}\) from \(\mathbf{r}\). This results in the vector \(\mathbf{r}-\mathbf{r}_{0} = \langle x-1, y-1, z-1 \rangle\).

Calculate the magnitude

Calculate the magnitude (length) of the vector from step 1 using the formula \(\sqrt{(x-1)^2 + (y-1)^2 + (z-1)^2}\). This is equivalent to finding the distance from the point \((x,y,z)\) to the point \((1,1,1)\).

Set the magnitude equal to the given value

Set the magnitude calculated in step 2 equal to 2 (the given length of the vector from \((1,1,1)\) to \((x,y,z)\)) to obtain the equation \((x-1)^2 + (y-1)^2 + (z-1)^2 = 2^2\).

Simplify the equation

Simplify the equation found in step 3 to its final form. Thus, we have the equation of a sphere of radius 2 centered at the point \((1,1,1)\) as \((x-1)^2 + (y-1)^2 + (z-1)^2 = 4\).