Final answer:
A vector can be represented as a linear combination of a given basis in three-dimensional space, such as  = Axî + Ayâ + Aê using the standard unit vectors. For a different basis, the vector is expressed in terms of new basis vectors and corresponding coordinates.
Step-by-step explanation:
The question seems to ask how to represent a vector with respect to a given basis which, based on the fragment provided ('basis 1 x³'), isn't entirely clear. However, in general, a vector can be represented in terms of a new basis by rewriting it as a linear combination of the basis vectors.
In the context of the provided information, the basis vectors for three-dimensional space are typically the unit vectors î (x-axis), â (y-axis), and ê (z-axis). A vector  in this space would be represented as  = Axî + Ayâ + Aê.
To exemplify representing a vector with respect to another basis, if we have a vector â and a new basis B = {b₁, b₂, b₃}, we express â as a1b₁ + a2b₂ + a3b₃, where a1, a2, and a3 are the coordinates of vector â with respect to basis B.