Final answer:
The coordinate vector of a point in the standard basis for ℝ² or ℝ³ is the point itself, represented as coordinates (x, y) or (x, y, z). When converting from polar or spherical coordinates to Cartesian, trigonometric relations are used to find the Cartesian coordinates.
Step-by-step explanation:
If we consider the standard basis for ℝ² or ℝ³ (the Cartesian coordinate system for the plane or space, respectively), then the question seems to be asking about the representation of a vector in the standard basis. In this basis, each vector is represented by its coordinates, which effectively means that the coordinate vector of a point is the point itself. For example, in ℝ², a vector ℝ = (x, y) is represented in the standard basis as the same tuple (x, y). Similarly, in ℝ³, a vector ℝ = (x, y, z) is represented as the tuple (x, y, z).
In contrast, when converting from polar coordinates to Cartesian coordinates, the relationships in two-dimensional space are x = r cos θ and y = r sin θ, where r is the magnitude of the vector and θ is the angle it makes with the x-axis. In three-dimensional space, converting from spherical coordinates involves similar trigonometric relationships: x = r sin ϕ cos θ, y = r sin ϕ sin θ, and z = r cos ϕ, where r is the radial distance, ϕ is the polar angle, and θ is the azimuthal angle.