# Obtain the vector and cartesian equation of thesphere whose centre is $(1 , -1 , 1 )$ and radius is the same as that of the sphere $|\overrightarrow{r}-(\overrightarrow{i}+\overrightarrow{j}+\overrightarrow{2k})|=5.$

Toolbox:
• Vector equation of a sphere with centre $\overrightarrow c$ and radius $a$ is $|\overrightarrow r-\overrightarrow c|=a$ When the centre is at the origin, the vector equation is $|\overrightarrow r|=a$ Cartesian equation is $(x-c_1)^2+(y-c_2)^2+(z-c_3)^2=a^2$ where $(c_1, c_2, c_3 )$ is the centre.
The sphere $|\overrightarrow r-(\overrightarrow i+\overrightarrow j+2\overrightarrow k)|=5$ is 5 units.
The required sphere has its centre at $(1, -1, 1)(\overrightarrow c=\overrightarrow i-\overrightarrow j+\overrightarrow k)$ and has a radius of 5 units.
The vector equation of the sphere is
$|\overrightarrow r-\overrightarrow c|=5$ i.e., $|\overrightarrow r-(\overrightarrow i-\overrightarrow j+\overrightarrow k)|=5$
The cartesian equation is
$(x-c_1)^2+(y-c_2)^2+(z-c_3)^2=a^2$
$(x-1)^2+(y+1)^2+(z-1)^2=25$
$x^2+y^2+z^2-2x+2y-2z+1+1+1-25=0$
$\therefore x^2+y^2+z^2-2x+2y-2z-22=0$