Statement 1: if three points `P ,Qa n dR` have position vectors ` vec a , vec b ,a n d vec c` , respectively, and `2 vec a+3 vec b-5 vec c=0,` then the points `P ,Q ,a n dR` must be collinear. Statement 2: If for three points `A ,B ,a n dC , vec A B=lambda vec A C ,` then points `A ,B ,a n dC` must be collinear.

Options

A. Both the statements are true, and Statement 2 is the correct explanation for Statement 1.
B. Both the statements are true, but Statement 2 is not the correct explanation for Statement 1.
C. Statement 1 is true and Statement 2 is false.
D. Statement 1 is false and Statement 2 is true.

Explanation

To solve the problem, we need to analyze both statements step by step.

### Step 1: Analyze Statement 1
We are given that the position vectors of points P, Q, and R are represented by vectors **a**, **b**, and **c** respectively. The equation provided is:

\[ 2\vec{a} + 3\vec{b} - 5\vec{c} = 0 \]

**Rearranging the equation:**
We can rearrange this equation to isolate the terms involving the position vectors:

\[ 3\vec{b} = 5\vec{c} - 2\vec{a} \]

### Step 2: Expressing in terms of vectors AB and AC
From the rearranged equation, we can express the relationship between the vectors **AB** and **AC**.

Recall that:
- **AB** = **b** - **a**
- **AC** = **c** - **a**

Now, substituting these into our rearranged equation:

\[ 3(\vec{b} - \vec{a}) = 5(\vec{c} - \vec{a}) \]

This simplifies to:

\[ 3\vec{AB} = 5\vec{AC} \]

### Step 3: Conclusion from the relationship
From the equation \( 3\vec{AB} = 5\vec{AC} \), we can express **AB** in terms of **AC**:

\[ \vec{AB} = \frac{5}{3}\vec{AC} \]

Since both vectors **AB** and **AC** share a common point (point A), this implies that points P, Q, and R must be collinear.

### Step 4: Analyze Statement 2
Statement 2 states that if for three points A, B, and C, we have:

\[ \vec{AB} = \lambda \vec{AC} \]

This means that the vector **AB** is a scalar multiple of the vector **AC**.

### Step 5: Conclusion from Statement 2
If one vector is a scalar multiple of another, it indicates that the two vectors are parallel. Since both vectors originate from point A, this confirms that points A, B, and C are collinear.

### Final Conclusion
Both statements are true, and statement 2 serves as a correct explanation for statement 1.

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