Two vectors, A and B, are added together to form the vector `C = A + B.` The realtionship between the magnitudes of these vectors is given by:
`C _(x) =0`
` C _(y) = A sin 60^(@) + B sin 30^(@)`
`A _(x) and A _(y)` point in the positive x and y directions, respectively.
Which one of the following statements best describes the orientation of vectors A and B ?

Options

A. A and B point in opposite directions.
B. A points `60^(@)` above the positive x axis while B points `30^(@)` above the negative x axis.
C. A point `60^(@)` above the negative x axis while B points `30^(@)` above the positive x axis.
D. A point `60^(@)` below the positive x axis while B points `30^(@)` above the positive y axis.

Answer

To solve the problem, we need to analyze the given information about the vectors A and B, their components, and the resultant vector C. Let's break it down step by step.

### Step 1: Understand the Components of Vectors
We know that the vectors A and B can be represented in terms of their components along the x and y axes:
- Vector A can be expressed as:
- \( A_x = A \cos(60^\circ) \)
- \( A_y = A \sin(60^\circ) \)

- Vector B can be expressed as:
- \( B_x = B \cos(30^\circ) \)
- \( B_y = B \sin(30^\circ) \)

### Step 2: Analyze the Given Information
From the problem, we have the following relationships:
- The x-component of vector C is given as \( C_x = 0 \).
- The y-component of vector C is given as \( C_y = A \sin(60^\circ) + B \sin(30^\circ) \).

### Step 3: Set Up the Equations
Since \( C_x = A_x + B_x = 0 \), we can write:
\[ A_x + B_x = 0 \]
This implies:
\[ A \cos(60^\circ) + B \cos(30^\circ) = 0 \]

### Step 4: Calculate the Cosine Values
Using the known values:
- \( \cos(60^\circ) = \frac{1}{2} \)
- \( \cos(30^\circ) = \frac{\sqrt{3}}{2} \)

Substituting these values into the equation gives:
\[ A \cdot \frac{1}{2} + B \cdot \frac{\sqrt{3}}{2} = 0 \]
This simplifies to:
\[ A + B\sqrt{3} = 0 \]
Thus, we can express \( A \) in terms of \( B \):
\[ A = -B\sqrt{3} \]

### Step 5: Analyze the Y-component
Now, for the y-component:
\[ C_y = A \sin(60^\circ) + B \sin(30^\circ) \]
Using the sine values:
- \( \sin(60^\circ) = \frac{\sqrt{3}}{2} \)
- \( \sin(30^\circ) = \frac{1}{2} \)

Substituting these values gives:
\[ C_y = A \cdot \frac{\sqrt{3}}{2} + B \cdot \frac{1}{2} \]

### Step 6: Substitute A into the Y-component Equation
Substituting \( A = -B\sqrt{3} \) into the y-component equation:
\[ C_y = (-B\sqrt{3}) \cdot \frac{\sqrt{3}}{2} + B \cdot \frac{1}{2} \]
This simplifies to:
\[ C_y = -\frac{3B}{2} + \frac{B}{2} = -B \]

### Step 7: Conclusion about the Orientation
From the analysis:
- Vector A points 60 degrees above the positive x-axis.
- Vector B points 30 degrees above the negative x-axis (which is equivalent to 180° - 30° = 150° from the positive x-axis).

### Final Answer
Thus, the orientation of the vectors is:
- Vector A is at 60 degrees from the positive x-axis.
- Vector B is at 30 degrees from the negative x-axis.