Final answer:
The coordinates of the point with respect to the new basis i and j will remain the same as the original coordinates with respect to the standard basis i and j.
Step-by-step explanation:
In a rectangular (Cartesian) xy-coordinate system in a plane, a point is described by a pair of coordinates (x, y). In the given question, the point is described with respect to the standard basis vectors i and j. If we draw the xy-plane with new basis vectors u and v, the coordinates of the point with respect to this new basis can be found as a combination of the new basis vectors.
a) u and v
Answer: The coordinates of the point with respect to the new basis u and v are found by expressing the point as a linear combination of u and v.
b) i and u
Answer: Since the question states that the new basis vectors are u and v, using i and u as the basis vectors would not correspond to the new basis and hence would not give the correct coordinates.
c) j and v
Answer: Using j and v as the basis vectors would not correspond to the new basis and hence would not give the correct coordinates.
d) i and j
Answer: The coordinates of the point with respect to the new basis i and j will remain the same as the original coordinates with respect to the standard basis i and j. Therefore, the correct choice for the new basis vectors is option d) i and j.