The position vectors of the points P and Q with respect to the origin O are `veca = hati + 3hatj-2hatk and vecb = 3hati -hatj -2hatk`, respectively. If M is a point on PQ, such that OM is the bisector of POQ, then `vec(OM)` is

Options

A. `2(hati-hatj+hatk)`
B. `2 hati+hatj-2hatk`
C. `2(-hati+hatj-hatk)`
D. `2(hati+hatj+hatk)`

Explanation

To solve the problem step by step, we will find the position vector of point M, which is the bisector of the angle formed by the vectors from O to P and from O to Q.

### Step 1: Identify the position vectors of points P and Q
Given:
- The position vector of point P is \(\vec{a} = \hat{i} + 3\hat{j} - 2\hat{k}\)
- The position vector of point Q is \(\vec{b} = 3\hat{i} - \hat{j} - 2\hat{k}\)

### Step 2: Calculate the position vector of point M
Since M is the point on line segment PQ such that OM is the bisector of angle POQ, M will be the midpoint of PQ.

The formula for the midpoint M of two points P and Q is given by:
\[ \vec{OM} = \frac{\vec{OP} + \vec{OQ}}{2} \]

### Step 3: Substitute the position vectors into the midpoint formula
Substituting \(\vec{a}\) and \(\vec{b}\) into the formula:
\[ \vec{OM} = \frac{\vec{a} + \vec{b}}{2} \]
\[ \vec{OM} = \frac{(\hat{i} + 3\hat{j} - 2\hat{k}) + (3\hat{i} - \hat{j} - 2\hat{k})}{2} \]

### Step 4: Combine the vectors
Now, we combine the vectors:
\[ \vec{OM} = \frac{(\hat{i} + 3\hat{j} - 2\hat{k} + 3\hat{i} - \hat{j} - 2\hat{k})}{2} \]
Combine like terms:
\[ = \frac{(1 + 3)\hat{i} + (3 - 1)\hat{j} + (-2 - 2)\hat{k}}{2} \]
\[ = \frac{4\hat{i} + 2\hat{j} - 4\hat{k}}{2} \]

### Step 5: Simplify the expression
Now, simplify the expression:
\[ \vec{OM} = 2\hat{i} + \hat{j} - 2\hat{k} \]

### Final Answer
Thus, the position vector of point M is:
\[ \vec{OM} = 2\hat{i} + \hat{j} - 2\hat{k} \]