Three vectors A, B, and C add together to yield zero : `A + B + C =0.` The vectors A and C point in opposite direction and their magnitudes are related by the expression: ` A = 2C.` Which one of the following conclusions is correct ?

Options

A. A and B have equal magnitudes and point in opposite directions.
B. B and C have equsal magnitudes and point in the same direction.
C. B and C have equal magnitudes and point in opposite directions.
D. A and B point in the same direction, but A has twice the magnitude of B.

Answer

To solve the problem, we need to analyze the given vectors A, B, and C based on the information provided. Let's break down the solution step-by-step.

### Step 1: Understand the Equation
We know that the vectors A, B, and C add up to zero:
\[ A + B + C = 0 \]
This can be rearranged to express B in terms of A and C:
\[ B = - (A + C) \]

### Step 2: Analyze the Directions of Vectors A and C
We are told that vectors A and C point in opposite directions. This means if we assume vector A points in the positive x-direction, then vector C must point in the negative x-direction.

### Step 3: Relate the Magnitudes of A and C
We are given the relationship between the magnitudes of vectors A and C:
\[ |A| = 2|C| \]
Let’s denote the magnitude of vector C as \( |C| = x \). Therefore, the magnitude of vector A will be:
\[ |A| = 2x \]

### Step 4: Assign Directions and Magnitudes
Assuming vector C points in the negative x-direction:
\[ C = -xi \]
Since A is in the opposite direction and has twice the magnitude:
\[ A = 2xi \]

### Step 5: Substitute A and C into the Equation for B
Now we can substitute A and C into the equation for B:
\[ B = - (A + C) \]
\[ B = - (2xi - xi) \]
\[ B = - (2xi - xi) = -xi \]

### Step 6: Analyze the Magnitudes and Directions of B and C
From our calculations:
- The magnitude of vector B is \( |B| = x \) (since \( B = -xi \)).
- The magnitude of vector C is also \( |C| = x \).
- Both vectors B and C point in the same direction (negative x-direction).

### Conclusion
From the analysis, we conclude that:
- Vectors B and C have equal magnitudes and point in the same direction.

### Final Answer
The correct conclusion is that **B and C are equal vectors**.