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STATEMENT-1 : If veca and vecb be two gi...

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.

A

F T T

B

T T F

C

F T F

D

T T T

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AI Generated Solution

The correct Answer is:
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.
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AAKASH INSTITUTE ENGLISH-VECTOR ALGEBRA-SECTION-H (Multiple True-Flase Type Questions)
  1. STATEMENT-1 : If veca and vecb be two given non zero vectors then any ...

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