Question

If r = x, y, z and r0 = x0, y0, z0 , describe the set of all points (x, y, z) such that |r − r0| = 5.

Answer 1

We are concern about the vector (x-x0, y-y0, z-z0) equaling to 5.

This happens when √(x-x0)^2 + (y-y0)^2 + (z-z0)^2 = 5

Square both side:

(x-x0)^2 + (y-y0)^2 + (z-z0)^2 = 25 , a sphere

Which you will recognize as a circle of radius one centered at (x0, y0, z0)

 

Answer 2

Answer:

Point $\left ( x,y,z\right )$ represents a sphere

Step-by-step explanation:

Take $r=\left ( x,y,z \right )\,,\,r_0=\left ( x_0,y_0,z_0 \right )$

We need to describe the set of all points (x, y, z) such that $\left | r-r_0 \right |=5$

Solution :

For $r-r_0=\left ( x,y,z \right )-\left ( x_0,y_0,z_0 \right )$, we will subtract the respective elements .

$r-r_0=\left ( x,y,z \right )-\left ( x_0,y_0,z_0 \right )=\left ( x-x_0,y-y_0,z-z_0 \right )$

Therefore, $\left | r-r_0 \right |=\sqrt{(x-x_0)^2+(y-y_0)^2+(z-z_0)^2}$

As $\left | r-r_0 \right |=5$, we get

$\sqrt{(x-x_0)^2+(y-y_0)^2+(z-z_0)^2}=5$

On squaring both sides, we get

$\left ( \sqrt{(x-x_0)^2+(y-y_0)^2+(z-z_0)^2} \right )^2=5^2\\(x-x_0)^2+(y-y_0)^2+(z-z_0)^2=25$

The general equation of a sphere is (x - a)² + (y - b)² + (z - c)² = r², where (a, b, c) denotes the center of the sphere and r represents the radius .

So, $(x-x_0)^2+(y-y_0)^2+(z-z_0)^2=25$ represents a sphere with center as $\left ( x_0,y_0,z_0 \right )$ and radius equal to 5 units