Let the position vectors of vertices `A,B,C` of `DeltaABC` be respectively `veca,vecb` and `vecc`. If `vecr` is the position vector of the mid point of the line segment joining its orthocentre and centroid then `(veca-vecr)+(vecb-vecr)+(vecc-vecr)=`

Options

A. A. 1
B. B. 2
C. C. 3
D. D. none of these

Explanation

To solve the problem step by step, we need to find the expression \((\vec{a} - \vec{r}) + (\vec{b} - \vec{r}) + (\vec{c} - \vec{r})\), where \(\vec{r}\) is the position vector of the midpoint of the line segment joining the orthocenter and centroid of triangle \(ABC\).

### Step 1: Understand the Position Vectors
Let the position vectors of the vertices of triangle \(ABC\) be:
- \(\vec{A} = \vec{a}\)
- \(\vec{B} = \vec{b}\)
- \(\vec{C} = \vec{c}\)

### Step 2: Find the Centroid \(\vec{G}\)
The centroid \(\vec{G}\) of triangle \(ABC\) is given by the formula:
\[ \vec{G} = \frac{\vec{a} + \vec{b} + \vec{c}}{3} \]

### Step 3: Find the Orthocenter \(\vec{H}\)
The orthocenter \(\vec{H}\) is a point that can be calculated using the properties of the triangle, but for this problem, we will use the fact that we need the midpoint of the segment joining \(\vec{G}\) and \(\vec{H}\).

### Step 4: Find the Midpoint \(\vec{r}\)
The position vector \(\vec{r}\) of the midpoint of the line segment joining the orthocenter \(\vec{H}\) and the centroid \(\vec{G}\) is given by:
\[ \vec{r} = \frac{\vec{G} + \vec{H}}{2} \]

### Step 5: Substitute \(\vec{G}\) in \(\vec{r}\)
Substituting \(\vec{G}\) into the equation for \(\vec{r}\):
\[ \vec{r} = \frac{\frac{\vec{a} + \vec{b} + \vec{c}}{3} + \vec{H}}{2} \]

### Step 6: Calculate the Expression
Now we need to compute:
\[ (\vec{a} - \vec{r}) + (\vec{b} - \vec{r}) + (\vec{c} - \vec{r}) \]
This can be rewritten as:
\[ (\vec{a} + \vec{b} + \vec{c}) - 3\vec{r} \]

### Step 7: Substitute \(\vec{r}\) into the Expression
Substituting \(\vec{r}\) into the expression gives:
\[ (\vec{a} + \vec{b} + \vec{c}) - 3\left(\frac{\frac{\vec{a} + \vec{b} + \vec{c}}{3} + \vec{H}}{2}\right) \]
This simplifies to:
\[ (\vec{a} + \vec{b} + \vec{c}) - \frac{\vec{a} + \vec{b} + \vec{c}}{2} - \frac{3\vec{H}}{2} \]

### Step 8: Combine Like Terms
Combining the terms results in:
\[ \frac{2(\vec{a} + \vec{b} + \vec{c})}{2} - \frac{\vec{a} + \vec{b} + \vec{c}}{2} - \frac{3\vec{H}}{2} = \frac{\vec{a} + \vec{b} + \vec{c}}{2} - \frac{3\vec{H}}{2} \]

### Step 9: Final Expression
Thus, the final expression simplifies to:
\[ \frac{\vec{a} + \vec{b} + \vec{c} - 3\vec{H}}{2} \]

### Conclusion
The final answer is:
\[ (\vec{a} - \vec{r}) + (\vec{b} - \vec{r}) + (\vec{c} - \vec{r}) = \frac{\vec{a} + \vec{b} + \vec{c} - 3\vec{H}}{2} \]