Question

Decompose vector (v) into two vectors (V_1) and (V_2), where (V_1) is parallel to (w) and (V_2) is orthogonal to (w).

A. (V_1 = i + 9j), (V_2 = 0)
B. (V_1 = 0), (V_2 = i + 9j)
C. (V_1 = i + j), (V_2 = 9i - j)
D. (V_1 = 9i - j), (V_2 = i + j)

Answer 1

To properly decompose a vector into parallel and orthogonal components relative to another vector, the direction and magnitude of that second vector are required, which are not provided in the student's question, and thus it's not possible to select the correct answer from the given options.

Step-by-step explanation:

To decompose a vector v into two components: one parallel and the other orthogonal to a given vector w, we must project v onto w to obtain the parallel component and subtract this projection from v to get the orthogonal component. If we're given vector v and a direction vector w, the component of v parallel to w, V1, is found by:

1. Finding the dot product of v and w.
2. Dividing the dot product by the magnitude squared of w.
3. Multiplying w by the resulting scalar to scale it to the parallel component.

After determining V1, we calculate the orthogonal component, V2, by:

1. Subtracting the parallel component V1 from v.

Given the provided answer choices and the vector w, we could verify which option correctly represents V1 and V2. However, since vector w has not been specified in the student question, we don't have enough information to definitively select the correct decomposition from options A, B, C, or D.