If a, b, c are non-collinear vectors such that `a + b` is parallel to c, and c + a is parallel to b, then

Options

A. `a + b = c`
B. a, b,c taken in order, form the sides of a triangle
C. `b + c =a`
D. none of these

Answer

To solve the problem, we need to analyze the given conditions about the vectors \( \vec{a}, \vec{b}, \vec{c} \) and their relationships.

### Step-by-Step Solution:

1. **Understanding the Conditions**:
We have two conditions:
- \( \vec{a} + \vec{b} \) is parallel to \( \vec{c} \).
- \( \vec{c} + \vec{a} \) is parallel to \( \vec{b} \).

2. **Expressing Parallel Vectors**:
If two vectors are parallel, it means one is a scalar multiple of the other. Therefore, we can write:
\[ \vec{c} = k_1 (\vec{a} + \vec{b}) \quad \text{for some scalar } k_1 \]
\[ \vec{b} = k_2 (\vec{c} + \vec{a}) \quad \text{for some scalar } k_2 \]

3. **Substituting the First Condition into the Second**:
Substitute \( \vec{c} \) from the first equation into the second:
\[ \vec{b} = k_2 (k_1 (\vec{a} + \vec{b}) + \vec{a}) \]
Simplifying this gives:
\[ \vec{b} = k_2 (k_1 \vec{a} + k_1 \vec{b} + \vec{a}) = k_2 ((k_1 + 1) \vec{a} + k_1 \vec{b}) \]

4. **Rearranging the Equation**:
Rearranging the equation gives:
\[ \vec{b} - k_2 k_1 \vec{b} = k_2 (k_1 + 1) \vec{a} \]
This can be factored as:
\[ (1 - k_2 k_1) \vec{b} = k_2 (k_1 + 1) \vec{a} \]

5. **Analyzing the Result**:
Since \( \vec{a}, \vec{b}, \vec{c} \) are non-collinear, the coefficients of \( \vec{a} \) and \( \vec{b} \) must equal zero for the equality to hold. This leads to two equations:
- \( 1 - k_2 k_1 = 0 \)
- \( k_2 (k_1 + 1) = 0 \)

6. **Solving the Equations**:
From \( 1 - k_2 k_1 = 0 \), we have:
\[ k_2 k_1 = 1 \quad \text{(1)} \]
From \( k_2 (k_1 + 1) = 0 \), since \( k_2 \neq 0 \) (otherwise \( \vec{b} \) would be zero), we must have:
\[ k_1 + 1 = 0 \Rightarrow k_1 = -1 \quad \text{(2)} \]

7. **Substituting Back**:
Substitute \( k_1 = -1 \) into equation (1):
\[ k_2 (-1) = 1 \Rightarrow k_2 = -1 \]

8. **Conclusion**:
Thus, we have:
\[ \vec{c} = -(\vec{a} + \vec{b}) \quad \text{and} \quad \vec{b} = -(\vec{c} + \vec{a}) \]
This implies that the vectors \( \vec{a}, \vec{b}, \vec{c} \) form a closed triangle.

### Final Answer:
The correct conclusion is that the vectors \( \vec{a}, \vec{b}, \vec{c} \) taken in order form the sides of a triangle.