If a and b are two non collinear vectors; then every vector r coplanar with a and b can be expressed in one and only one way as a linear combination: xa+yb=0.

Options

A. (a)x=0, but y is not necessarily zero
B. (b)y=0, but x is not necessarily zero
C. (c)x=0,y=0
D. (d)none of these

Explanation

To solve the problem, we need to show that if \( \mathbf{a} \) and \( \mathbf{b} \) are two non-collinear vectors, then every vector \( \mathbf{r} \) that is coplanar with \( \mathbf{a} \) and \( \mathbf{b} \) can be expressed in one and only one way as a linear combination of \( \mathbf{a} \) and \( \mathbf{b} \).

### Step-by-Step Solution:

1. **Understanding Coplanarity**:
- Two vectors \( \mathbf{a} \) and \( \mathbf{b} \) are non-collinear, meaning they do not lie on the same line. Any vector \( \mathbf{r} \) that is coplanar with \( \mathbf{a} \) and \( \mathbf{b} \) can be expressed as a linear combination of these vectors.

2. **Linear Combination**:
- A vector \( \mathbf{r} \) can be expressed as:
\[ \mathbf{r} = x \mathbf{a} + y \mathbf{b} \]
- Here, \( x \) and \( y \) are scalars.

3. **Setting Up the Equation**:
- According to the problem statement, we need to show that:
\[ x \mathbf{a} + y \mathbf{b} = \mathbf{0} \]
- This implies that the vector \( \mathbf{r} \) is the zero vector.

4. **Rearranging the Equation**:
- Rearranging gives:
\[ x \mathbf{a} = -y \mathbf{b} \]

5. **Analyzing the Implications**:
- If we assume \( x \neq 0 \) and \( y \neq 0 \), then we can express \( \mathbf{a} \) in terms of \( \mathbf{b} \) (or vice versa), which would imply that \( \mathbf{a} \) and \( \mathbf{b} \) are collinear. This contradicts our initial condition that \( \mathbf{a} \) and \( \mathbf{b} \) are non-collinear.

6. **Conclusion**:
- Therefore, the only solution that does not contradict the non-collinearity of \( \mathbf{a} \) and \( \mathbf{b} \) is when both \( x \) and \( y \) are equal to zero:
\[ x = 0 \quad \text{and} \quad y = 0 \]
- This means that the only vector \( \mathbf{r} \) that can be expressed as a linear combination of \( \mathbf{a} \) and \( \mathbf{b} \) in this case is the zero vector.

7. **Final Answer**:
- Thus, the correct option is:
\[ \text{Option C: } x = 0 \text{ and } y = 0 \]