Question

Write the vector u=(-1,2,2) as the sum of a vector parallel to v=(2,-2,3) and a vector orthogonal to v.

Options:
a) u = (1, 0, 0) + (0, 2, 2)
b) u = (-1, 2, 2) + (0, 0, 0)
c) u = (1, 0, 0) + (0, -2, -3)
d) u = (0, 2, 2) + (-1, 0, 0)

Answer 1

To decompose a vector into parallel and orthogonal components relative to another vector, one would use projection formulas. However, none of the provided answer choices correctly represent the decomposition of vector u=(-1,2,2) relative to vector v=(2,-2,3). Therefore, none of the choices can be endorsed.

Step-by-step explanation:

The question involves writing the vector u=(-1,2,2) as the sum of a vector parallel to v=(2,-2,3) and a vector orthogonal to v. To find the vector parallel to v, we can project u onto v by using the following projection formula:

projv(u) = (u · v / |v|2) v

Next, we subtract this projection from u to obtain the vector orthogonal to v:

uorthogonal = u - projv(u)

We combine these two vectors to express u as the sum of the parallel and orthogonal components making sure that the sum indeed equals the original vector u. Nonetheless, among the provided options, none are the result of this process, but rather display different decompositions. In this case, we decline to endorse any of the choices because they do not correctly represent the mathematics described.