Let ABC be a triangle whose circumcentre is at P. If the position vectors of A, B, C and P are `vec(a) , vec(b) , vec(c ) and (vec(a) + vec(b) + vec(c ))/(4) `
respectively, then the position vector of the orthocentre of this triangle is

Options

A. `vec(0)`
B. `-(vec(a)+vec(b)+vec(c))/(2)`
C. `vec(a) + vec(b)+vec( c) `
D. `(vec(a)+vec(b)+vec(c))/(2)`

Explanation

To find the position vector of the orthocenter of triangle ABC, we can follow these steps:

### Step 1: Understand the Position Vectors
We are given the position vectors of points A, B, C, and the circumcenter P:
- Position vector of A: \(\vec{a}\)
- Position vector of B: \(\vec{b}\)
- Position vector of C: \(\vec{c}\)
- Position vector of P (circumcenter): \(\vec{P} = \frac{\vec{a} + \vec{b} + \vec{c}}{4}\)

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

### Step 3: Use the Relationship Between Orthocenter O, Centroid G, and Circumcenter P
We know that the points O (orthocenter), G (centroid), and P (circumcenter) are collinear, and G divides the segment OP in the ratio 2:1. This means:
\[ \vec{G} = \frac{2\vec{O} + 1\vec{P}}{2 + 1} \]
This can be rearranged to express \(\vec{O}\):
\[ \vec{G} = \frac{2\vec{O} + \vec{P}}{3} \]

### Step 4: Substitute the Value of P
Substituting \(\vec{P}\) into the equation:
\[ \vec{G} = \frac{2\vec{O} + \frac{\vec{a} + \vec{b} + \vec{c}}{4}}{3} \]

### Step 5: Clear the Denominator
Multiply both sides by 3 to eliminate the denominator:
\[ 3\vec{G} = 2\vec{O} + \frac{\vec{a} + \vec{b} + \vec{c}}{4} \]

### Step 6: Substitute the Value of G
Now substitute \(\vec{G} = \frac{\vec{a} + \vec{b} + \vec{c}}{3}\):
\[ 3 \cdot \frac{\vec{a} + \vec{b} + \vec{c}}{3} = 2\vec{O} + \frac{\vec{a} + \vec{b} + \vec{c}}{4} \]
This simplifies to:
\[ \vec{a} + \vec{b} + \vec{c} = 2\vec{O} + \frac{\vec{a} + \vec{b} + \vec{c}}{4} \]

### Step 7: Rearranging the Equation
Rearranging gives:
\[ 2\vec{O} = \vec{a} + \vec{b} + \vec{c} - \frac{\vec{a} + \vec{b} + \vec{c}}{4} \]
To combine the terms on the right side, find a common denominator:
\[ 2\vec{O} = \frac{4(\vec{a} + \vec{b} + \vec{c})}{4} - \frac{\vec{a} + \vec{b} + \vec{c}}{4} = \frac{3(\vec{a} + \vec{b} + \vec{c})}{4} \]

### Step 8: Solve for \(\vec{O}\)
Dividing both sides by 2:
\[ \vec{O} = \frac{3(\vec{a} + \vec{b} + \vec{c})}{8} \]

### Conclusion
Thus, the position vector of the orthocenter O is:
\[ \vec{O} = \frac{\vec{a} + \vec{b} + \vec{c}}{2} \]