In a triangle ABC, if taken in order, consider the following statements
1. `vec(AB) + vec(BC) + vec(CA) = vec(0)`
2 `vec(AB) + vec(BC) - vec(CA) = vec(0)`
3. `vec(AB)- vec(BC) + vec(CA) = vec(0)`
4. `vec(BA)- vec(BC) + vec(CA) = vec(0)`
How many of the above statements are correct?

Options

A. one
B. two
C. Three
D. Four

Answer

To solve the problem, we will analyze each of the given statements regarding the vectors in triangle ABC.

### Step 1: Understand the Vectors in Triangle ABC
In triangle ABC, we can define the vectors as follows:
- \(\vec{AB} = \vec{B} - \vec{A}\)
- \(\vec{BC} = \vec{C} - \vec{B}\)
- \(\vec{CA} = \vec{A} - \vec{C}\)

### Step 2: Analyze Each Statement

**Statement 1:** \(\vec{AB} + \vec{BC} + \vec{CA} = \vec{0}\)

Substituting the vectors:
\[ \vec{AB} + \vec{BC} + \vec{CA} = (\vec{B} - \vec{A}) + (\vec{C} - \vec{B}) + (\vec{A} - \vec{C}) \]
Simplifying:
\[ = \vec{B} - \vec{A} + \vec{C} - \vec{B} + \vec{A} - \vec{C} = \vec{0} \]
**This statement is correct.**

**Statement 2:** \(\vec{AB} + \vec{BC} - \vec{CA} = \vec{0}\)

Substituting the vectors:
\[ \vec{AB} + \vec{BC} - \vec{CA} = (\vec{B} - \vec{A}) + (\vec{C} - \vec{B}) - (\vec{A} - \vec{C}) \]
Simplifying:
\[ = \vec{B} - \vec{A} + \vec{C} - \vec{B} - \vec{A} + \vec{C} = 2\vec{C} - 2\vec{A} \neq \vec{0} \]
**This statement is incorrect.**

**Statement 3:** \(\vec{AB} - \vec{BC} + \vec{CA} = \vec{0}\)

Substituting the vectors:
\[ \vec{AB} - \vec{BC} + \vec{CA} = (\vec{B} - \vec{A}) - (\vec{C} - \vec{B}) + (\vec{A} - \vec{C}) \]
Simplifying:
\[ = \vec{B} - \vec{A} - \vec{C} + \vec{B} + \vec{A} - \vec{C} = 2\vec{B} - 2\vec{C} \neq \vec{0} \]
**This statement is incorrect.**

**Statement 4:** \(\vec{BA} - \vec{BC} + \vec{CA} = \vec{0}\)

Substituting the vectors:
\[ \vec{BA} - \vec{BC} + \vec{CA} = (\vec{A} - \vec{B}) - (\vec{C} - \vec{B}) + (\vec{A} - \vec{C}) \]
Simplifying:
\[ = \vec{A} - \vec{B} - \vec{C} + \vec{B} + \vec{A} - \vec{C} = 2\vec{A} - 2\vec{C} \neq \vec{0} \]
**This statement is incorrect.**

### Conclusion
Only Statement 1 is correct. Therefore, the number of correct statements is **1**.