Two vectors A and B, are added together to form the vector `C = A+B.` The relationship between the magnitudes of these vectors is given by `C _(x) = A cos 30^(@)+B and C _(y)=-A sin 30^(@).` Which statement best describes the orientation of these vectors ?

Options

A. A points in the negative x direction while B points in the positive y direction.
B. A points in the negative y direction while B points in the positive x direction.
C. A points `30^(@)` below the positive x axis while B points in the positive x direction.
D. A points `30^(@)` above the positive x axis while B points in the positive x direction.

Answer

To solve the problem, we need to analyze the given relationships between the components of the vectors A, B, and C. The relationships are:

1. \( C_x = A \cos(30^\circ) + B \)
2. \( C_y = -A \sin(30^\circ) \)

### Step 1: Understand the Components
The vector \( C \) is the resultant of vectors \( A \) and \( B \). The components \( C_x \) and \( C_y \) represent the horizontal and vertical components of vector \( C \), respectively.

### Step 2: Analyze the Components
From the equations:
- The \( C_x \) component includes the positive contributions from both \( A \) and \( B \).
- The \( C_y \) component is negative, indicating that the resultant vector \( C \) has a downward direction in the vertical axis.

### Step 3: Determine the Orientation of Vector A
The component \( C_y = -A \sin(30^\circ) \) suggests that vector \( A \) has a vertical component acting in the negative direction. Since \( A \) has a component in the negative \( y \)-direction, it implies that vector \( A \) is directed downward.

### Step 4: Determine the Angle of Vector A
The angle of vector \( A \) with respect to the positive x-axis is \( 30^\circ \) below the x-axis. This means that vector \( A \) is oriented in the fourth quadrant.

### Step 5: Determine the Orientation of Vector B
The equation \( C_x = A \cos(30^\circ) + B \) indicates that vector \( B \) is acting along the positive x-direction. Since there is no vertical component from \( B \) (as indicated by the absence of a \( B \sin(\theta) \) term), vector \( B \) must be horizontal.

### Conclusion
Based on the analysis:
- Vector \( A \) is oriented \( 30^\circ \) below the positive x-axis (in the fourth quadrant).
- Vector \( B \) is oriented along the positive x-axis.

Thus, the best statement that describes the orientation of these vectors is:
**Vector A points \( 30^\circ \) below the positive x-axis, while vector B points in the positive x-direction.**