Final answer:
Two coordinate systems defined by the same basis vectors 'i' and 'j' are equivalent, and vector operations within these systems would yield the same components and results for any given vector.
Step-by-step explanation:
If we assume that two coordinate systems are defined by the same basis vectors i and j, it implies that both systems are equivalent and have the same unit vectors for their respective axes. In other words, their corresponding axis directions are identical; the i vector corresponds to the positive direction along the x-axis, and the j vector corresponds to the positive direction along the y-axis. This means that when you project any vector onto their axes, you would obtain the same components in both coordinate systems.
This scenario aligns with the basic principles of vector operations in a Cartesian coordinate system. The dot product of a vector with the unit vectors yields the respective component of that vector along the corresponding axis. Faultlessly, when we project a vector onto the basis vectors i and j in the proposed system, we would derive the x-component and y-component of the vector, which are scalar values representing the magnitude of the projection of the vector along the x-axis and y-axis, respectively.
Furthermore, the cross product of these unit vectors, as represented by i x j, yields a new vector that is orthogonal to the original plane, which for a standard three-dimensional Cartesian system would correspond to the unit vector k along the z-axis. Therefore, if two coordinate systems share the same unit vectors i and j, their spatial orientation and vector manipulation rules regarding addition, scalar multiplication, dot product, and cross product will be identical for a given vector expressed within these systems.