Statement 1: In `"Delta"A B C , vec A B+ vec A B+ vec C A=0` Statement 2: If ` vec O A= vec a , vec O B= vec b ,t h e n vec A B= vec a+ vec b`

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 question, we need to analyze both statements regarding vectors in a triangle.

### Step-by-Step Solution:

**Step 1: Analyze Statement 1**

In triangle \( ABC \), we are given the vector equation:
\[ \vec{AB} + \vec{BC} + \vec{CA} = 0 \]
This statement is based on the triangle law of vector addition. According to this law, if you take the vectors representing the sides of a triangle in order, their sum will equal zero because they form a closed loop.

**Conclusion for Statement 1:**
This statement is **true**.

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**Step 2: Analyze Statement 2**

We are given:
\[ \vec{OA} = \vec{a}, \quad \vec{OB} = \vec{b} \]
We need to find \( \vec{AB} \).

Using the vector subtraction formula:
\[ \vec{AB} = \vec{OB} - \vec{OA} \]
Substituting the values we have:
\[ \vec{AB} = \vec{b} - \vec{a} \]

However, the statement claims:
\[ \vec{AB} = \vec{a} + \vec{b} \]
This is incorrect, as we have derived that:
\[ \vec{AB} = \vec{b} - \vec{a} \]

**Conclusion for Statement 2:**
This statement is **false**.

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### Final Conclusion:
- Statement 1 is true.
- Statement 2 is false.

Therefore, the answer is **C**.

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