STATEMENT-1 : If `veca` and `vecb` be two given non zero vectors then any vector `vecr` coplanar with `veca` and `vecb` can be represented as `vecr=xveca+yvecb` where x &yare some scalars.
STATEMENT-2 : If `veca,vecbvecc` are three non-coplanar vectors, then any vector `vecr` in space can be expressed as `vecr=xveca+yvecb+zvecc`, where x,y,z are some scalars.
STATEMENT-3 : If vectors `veca & vecb` represent two sides of a triangle, then `lambda(veca+vecb)` (where `lambda ne 1`) can represent the vector along third side.

Options

A. F T T
B. T T F
C. F T F
D. T T T

Answer

To analyze the given statements, we will evaluate each statement one by one.

### Statement 1:
**Statement:** If \(\vec{a}\) and \(\vec{b}\) are two given non-zero vectors, then any vector \(\vec{r}\) coplanar with \(\vec{a}\) and \(\vec{b}\) can be represented as \(\vec{r} = x\vec{a} + y\vec{b}\) where \(x\) and \(y\) are some scalars.

**Solution:**
1. **Understanding Coplanarity:** Two vectors \(\vec{a}\) and \(\vec{b}\) define a plane. Any vector \(\vec{r}\) that lies in this plane can be expressed as a linear combination of \(\vec{a}\) and \(\vec{b}\).
2. **Linear Combination:** The expression \(\vec{r} = x\vec{a} + y\vec{b}\) represents a linear combination of the vectors \(\vec{a}\) and \(\vec{b}\). Since \(\vec{r}\) is coplanar with \(\vec{a}\) and \(\vec{b}\), this representation is valid.
3. **Conclusion:** Therefore, Statement 1 is **True**.

### Statement 2:
**Statement:** If \(\vec{a}, \vec{b}, \vec{c}\) are three non-coplanar vectors, then any vector \(\vec{r}\) in space can be expressed as \(\vec{r} = x\vec{a} + y\vec{b} + z\vec{c}\), where \(x, y, z\) are some scalars.

**Solution:**
1. **Understanding Non-Coplanarity:** Three non-coplanar vectors span a three-dimensional space. This means any vector in that space can be represented as a linear combination of these three vectors.
2. **Linear Combination:** The expression \(\vec{r} = x\vec{a} + y\vec{b} + z\vec{c}\) indicates that \(\vec{r}\) can be formed by scaling and adding the vectors \(\vec{a}, \vec{b},\) and \(\vec{c}\).
3. **Conclusion:** Therefore, Statement 2 is **True**.

### Statement 3:
**Statement:** If vectors \(\vec{a}\) and \(\vec{b}\) represent two sides of a triangle, then \(\lambda(\vec{a} + \vec{b})\) (where \(\lambda \neq 1\)) can represent the vector along the third side.

**Solution:**
1. **Understanding Triangle Sides:** In a triangle, if \(\vec{a}\) and \(\vec{b}\) are two sides, the third side can be represented as \(\vec{c} = \vec{b} - \vec{a}\) (assuming \(\vec{a}\) and \(\vec{b}\) are positioned tail-to-tail).
2. **Scaling Issue:** The expression \(\lambda(\vec{a} + \vec{b})\) does not necessarily represent the third side unless \(\lambda = 1\). For other values of \(\lambda\), it represents a vector that is not aligned with the third side.
3. **Conclusion:** Therefore, Statement 3 is **False**.

### Final Conclusion:
- Statement 1: True
- Statement 2: True
- Statement 3: False

### Summary:
- **Correctness of Statements:** Statement 1 is True, Statement 2 is True, and Statement 3 is False.