### Question 5If \( \mathbf{r} = \langle x, y, z \rangle \) and \( \mathbf{r_0} = \langle x_0, y_0, z_0 \rangle \), describe precisely the set of all points \( (x, y, z) \) such that\[ |\mathbf{r} - \mathbf{r_0}| = 1 \]### Answer ExplanationThis question asks you to describe the set of all points \((x, y, z)\) that are at a distance of 1 unit from a given point \((x_0, y_0, z_0)\). This set of points forms a sphere with radius 1 centered at the point \((x_0, y_0, z_0)\).In mathematical terms, the distance between any point \((x, y, z)\) and a fixed point \((x_0, y_0, z_0)\) in three-dimensional space is given by:\[ ((x - x_0)^2 + (y - y_0)^2 + (z - z_0)^2)^{1/2} \]For the distance to be equal to 1,\[ \sqrt{(x - x_0)^2 + (y - y_0)^2 + (z - z_0)^2} = 1 \]Squaring both sides, we get:\[ (x - x_0)^2 + (y - y_0)^2 + (z - z_0)^2 = 1 \]Thus, the set of all points \( (x, y, z) \) that satisfy this equation forms a sphere with radius 1 centered at \((x_0, y_0, z_0)\).This question illustrates the concept of the equation of a sphere and is fundamental in understanding the geometry of spheres in three-dimensional space.

Name: ### Question 5If \( \mathbf{r} = \langle x, y, z \rangle \) and \( \m
Uploaded: 2024-03-20T06:47:08.000Z
Channel: Bartleby
Description: ### Question 5If \( \mathbf{r} = \langle x, y, z \rangle \) and \( \mathbf{r_0} = \langle x_0, y_0, z_0 \rangle \), describe precisely the set of all points \( (x, y, z) \) such that\[ |\mathbf{r} - \mathbf{r_0}| = 1 \]### Answer ExplanationThis que