Let ABC be a triangle having its centroid its centroid at G. If S is any point in the plane of the triangle, then `vec(SA) + vec(SB)+vec(SC)=`

Options

A. Statement - 1 is True, Statement - 2 is True , Statement - 2 is a correct explanation for Statement - 1.
B. Statement -1 is True, Statement - 2 is True, Statement -2 is not a correct explanation for Statement - 1.
C. Statement - 1 is True, Statement - 2 is False.
D. Statement - 1 is False, Statement - 2 is True.

Explanation

To solve the problem, we need to find the vector sum of the vectors from point S to the vertices A, B, and C of triangle ABC. Let's denote the position vectors of points A, B, C, and S as \(\vec{A}\), \(\vec{B}\), \(\vec{C}\), and \(\vec{S}\) respectively.

### Step-by-Step Solution:

1. **Understanding the Centroid**:
The centroid \(G\) of triangle \(ABC\) is given by the formula:
\[ \vec{G} = \frac{\vec{A} + \vec{B} + \vec{C}}{3} \]

2. **Expressing Vectors from S to A, B, and C**:
We need to express the vectors \(\vec{SA}\), \(\vec{SB}\), and \(\vec{SC}\):
\[ \vec{SA} = \vec{A} - \vec{S}, \quad \vec{SB} = \vec{B} - \vec{S}, \quad \vec{SC} = \vec{C} - \vec{S} \]

3. **Summing the Vectors**:
Now, we can sum these vectors:
\[ \vec{SA} + \vec{SB} + \vec{SC} = (\vec{A} - \vec{S}) + (\vec{B} - \vec{S}) + (\vec{C} - \vec{S}) \]

4. **Simplifying the Expression**:
Combine the terms:
\[ \vec{SA} + \vec{SB} + \vec{SC} = \vec{A} + \vec{B} + \vec{C} - 3\vec{S} \]

5. **Substituting the Centroid**:
We know from the definition of the centroid that:
\[ \vec{A} + \vec{B} + \vec{C} = 3\vec{G} \]
Substituting this into our equation gives:
\[ \vec{SA} + \vec{SB} + \vec{SC} = 3\vec{G} - 3\vec{S} \]

6. **Factoring Out Common Terms**:
We can factor out the common term:
\[ \vec{SA} + \vec{SB} + \vec{SC} = 3(\vec{G} - \vec{S}) \]

### Final Result:
Thus, we conclude that:
\[ \vec{SA} + \vec{SB} + \vec{SC} = 3(\vec{G} - \vec{S}) \]