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# Find the vector equation of a sphere with centre having position vector $\overrightarrow{2i}-\overrightarrow{j}+\overrightarrow{3k}$ and radius $4$ units. Also find the equation in cartesian form.

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• 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.
Centre of the sphere has $pv \: \overrightarrow c=2\overrightarrow i-\overrightarrow j+3\overrightarrow k$ radius = 4 units.
Vector equation : $|\overrightarrow r-\overrightarrow c|=4$
$|\overrightarrow r-(2\overrightarrow i-\overrightarrow j+3\overrightarrow k)|=4$
Cartesian equation : $(x-2)^2+(y+1)^2+(z-3)^2=16$
$x^2+y^2+z^2-4x+2y-6z=16-4-1-9$
$x^2+y^2+z^2-4x+2y-6z-2=0$