If A,B and C are the vertices of a triangle with position vectors `vec(a) , vec(b) ` and `vec(c )` respectively and G is the centroid of `Delta ABC`, then `vec( GA) + vec( GB ) + vec( GC )` is equal to

Options

A. `vec( 0 ) `
B. `vec( a) + vec( b) + vec( c )`
C. `(vec( a) + vec( b) + vec( c ))/(3)`
D. `(vec( a) - vec( b) - vec( c ))/(3)`

Answer

To solve the problem, we need to find the expression for \( \vec{GA} + \vec{GB} + \vec{GC} \) where \( G \) is the centroid of triangle \( ABC \) with position vectors \( \vec{A}, \vec{B}, \vec{C} \).

### 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 G to A, B, and C**:
We can express the vectors \( \vec{GA}, \vec{GB}, \) and \( \vec{GC} \) in terms of \( \vec{G} \):
\[ \vec{GA} = \vec{A} - \vec{G} \]
\[ \vec{GB} = \vec{B} - \vec{G} \]
\[ \vec{GC} = \vec{C} - \vec{G} \]

3. **Adding the Vectors**:
Now, we can add these vectors together:
\[ \vec{GA} + \vec{GB} + \vec{GC} = (\vec{A} - \vec{G}) + (\vec{B} - \vec{G}) + (\vec{C} - \vec{G}) \]
Simplifying this, we get:
\[ = \vec{A} + \vec{B} + \vec{C} - 3\vec{G} \]

4. **Substituting the Value of G**:
Now substitute \( \vec{G} \) into the equation:
\[ = \vec{A} + \vec{B} + \vec{C} - 3\left(\frac{\vec{A} + \vec{B} + \vec{C}}{3}\right) \]
This simplifies to:
\[ = \vec{A} + \vec{B} + \vec{C} - (\vec{A} + \vec{B} + \vec{C}) \]
\[ = 0 \]

5. **Conclusion**:
Therefore, the final result is:
\[ \vec{GA} + \vec{GB} + \vec{GC} = \vec{0} \]

### Final Answer:
\[ \vec{GA} + \vec{GB} + \vec{GC} = \vec{0} \]