Find the Vector and Cartesian equation of the sphere whose centre $(2, - 3, 4)$ and radius is $5$ .

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Solution

Centre of the sphere $(2, - 3, 4)$ and radius of the sphere $a = 5$ Vector equation of sphere
$| \to r - \to c | = a$
$| \to r - (2^i - 3^j + 4^k) | = 5$
Cartesian equation
$(x - α)^{2} + (y - β)^{2} + (z - γ)^{2} = (a)^{2}$
$(x - 2)^{2} + (y + 3)^{2} + (z - 4)^{2} = 5^{2}$
$x^{2} + 4 - 4 x + y^{2} + 9 + 6 y + z^{2} + 16 - 8 z = 25$
$x^{2} + y^{2} + z^{2} - 4 x + 6 y - 8 z + 4 = 0$ .

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