If P, Q, R are the mid -points of the sides AB, BC, CA respectively of a `DeltaABC and vec(a).vec(p) vec(q)` are the position vectors of A.P.Q respectively, then what is the position vector of R?

Options

A. A. `2 vec(a) - (vec(p) -vec(q))`
B. B. `(vec(p)- vec(q))-2vec(a)`
C. C. `vec(a) - (vec(p) - vec(q))`
D. D. `(vec(a))/(2)- ((vec(p)- vec(q)))/(2)`

Answer

To find the position vector of point R, which is the midpoint of side AC in triangle ABC, we can follow these steps:

### Step 1: Define the Position Vectors
Let:
- \( \vec{A} \) be the position vector of point A.
- \( \vec{B} \) be the position vector of point B.
- \( \vec{C} \) be the position vector of point C.
- \( \vec{P} \) be the position vector of point P, the midpoint of AB.
- \( \vec{Q} \) be the position vector of point Q, the midpoint of BC.
- \( \vec{R} \) be the position vector of point R, the midpoint of CA.

### Step 2: Find the Position Vector of P
Since P is the midpoint of AB, we can express its position vector as:
\[ \vec{P} = \frac{\vec{A} + \vec{B}}{2} \]
Multiplying both sides by 2 gives:
\[ 2\vec{P} = \vec{A} + \vec{B} \quad \text{(Equation 1)} \]

### Step 3: Find the Position Vector of Q
Since Q is the midpoint of BC, we can express its position vector as:
\[ \vec{Q} = \frac{\vec{B} + \vec{C}}{2} \]
Multiplying both sides by 2 gives:
\[ 2\vec{Q} = \vec{B} + \vec{C} \quad \text{(Equation 2)} \]

### Step 4: Express B in terms of P and A
From Equation 1, we can rearrange to find \( \vec{B} \):
\[ \vec{B} = 2\vec{P} - \vec{A} \quad \text{(Equation 3)} \]

### Step 5: Substitute B in Equation 2
Now substitute Equation 3 into Equation 2 to find \( \vec{C} \):
\[ 2\vec{Q} = (2\vec{P} - \vec{A}) + \vec{C} \]
Rearranging gives:
\[ \vec{C} = 2\vec{Q} - (2\vec{P} - \vec{A}) = 2\vec{Q} - 2\vec{P} + \vec{A} \quad \text{(Equation 4)} \]

### Step 6: Find the Position Vector of R
Since R is the midpoint of AC, we can express its position vector as:
\[ \vec{R} = \frac{\vec{A} + \vec{C}}{2} \]
Substituting Equation 4 into this gives:
\[ \vec{R} = \frac{\vec{A} + (2\vec{Q} - 2\vec{P} + \vec{A})}{2} \]
Simplifying this:
\[ \vec{R} = \frac{2\vec{A} + 2\vec{Q} - 2\vec{P}}{2} = \vec{A} + \vec{Q} - \vec{P} \]

### Final Result
Thus, the position vector of R is:
\[ \vec{R} = \vec{A} + \vec{Q} - \vec{P} \]