Final answer:
The equation of the sphere in Euclidean coordinates is (x - 2)² + (y - 4)² + (z - 5)² = 9, and in vector form, it is represented by ||β - α|| = 3, where α is the center vector of the sphere.
Step-by-step explanation:
The equation of a sphere in Euclidean coordinates with center at (2, 4, 5) and radius 3 is (x - 2)² + (y - 4)² + (z - 5)² = 9. In vector form, the equation is written as ||β - α|| = 3, where α = <3, 2, 4, 5> is the center vector and β is the position vector of any point on the sphere.
An explanation of these equations is based on the standard formula for a sphere. In Euclidean geometry, the equation for a sphere centered at a point (h, k, l) with radius r is (x - h)² + (y - k)² + (z - l)² = r². Applying that to the given center (2, 4, 5) and radius 3, we achieve the direct answer above. The vector form replaces x, y, and z with a position vector β and equates its distance to the center vector α with the radius.