Cartesian coordinates in space

In space we take three mutually perpendicular axes Ox, Oy, Oz meeting at O; they determine the coordinate planes Oxy, Oyz, Oxz and divide space into 8 octants. Every point A is given by three numbers — the abscissa x, ordinate y and applicate z: A(x, y, z). Points on the Oz axis have the form (0, 0, c) and points of the plane Oxy have the form (x, y, 0). The distance between A(x₁, y₁, z₁) and B(x₂, y₂, z₂) is AB = √((x₂ − x₁)² + (y₂ − y₁)² + (z₂ − z₁)²) (the Pythagorean theorem applied twice). The midpoint of AB is ((x₁ + x₂)/2, (y₁ + y₂)/2, (z₁ + z₂)/2); the point P dividing AB in the ratio AP : PB = λ has coordinates (x₁ + λx₂)/(1 + λ), (y₁ + λy₂)/(1 + λ), (z₁ + λz₂)/(1 + λ). The sphere with centre (a, b, c) and radius R has the equation (x − a)² + (y − b)² + (z − c)² = R², and the ball is the inequality (x − a)² + (y − b)² + (z − c)² ≤ R². The distance from A(x, y, z) to the plane Oxy is |z|, and to the Ox axis √(y² + z²).