If C is the mid point of AB and P is any point outside AB then (A) `vec(PA)+vec(PB)+vec(PC)=0` (B) `vec(PA)+vec(PB)+2vec(PC)=vec0` (C) `vec(PA)+vec(PB)=vec(PC)` (D) `vec(PA)+vec(PB)=2vec(PC)`

Options

A. `P vec(A) + P vec(B) + P vec(C ) =vec(0)`
B. `P vec(A)+P vec(B) + 2P vec(C ) =vec(0)`
C. `P vec(A) +P vec(B)=P vec(C )`
D. `P vec(A) + P vec(B)=2P vec(C )`

Explanation

To solve the problem, we need to analyze the vectors involved and their relationships based on the given conditions. Let's denote the points as follows:

- Let \( A \) and \( B \) be two points in space.
- Let \( C \) be the midpoint of segment \( AB \).
- Let \( P \) be a point outside the line segment \( AB \).

We need to find the correct relationship among the vectors \( \vec{PA} \), \( \vec{PB} \), and \( \vec{PC} \).

### Step-by-step Solution:

1. **Identify the Midpoint**:
Since \( C \) is the midpoint of \( AB \), we can express the position vector of \( C \) as:
\[ \vec{C} = \frac{\vec{A} + \vec{B}}{2} \]

2. **Express the Vectors**:
The vectors from point \( P \) to points \( A \), \( B \), and \( C \) can be expressed as:
\[ \vec{PA} = \vec{A} - \vec{P} \]
\[ \vec{PB} = \vec{B} - \vec{P} \]
\[ \vec{PC} = \vec{C} - \vec{P} \]

3. **Using Triangle Law of Addition**:
In triangle \( PAC \):
\[ \vec{PA} + \vec{AC} = \vec{PC} \]
Here, \( \vec{AC} = \vec{C} - \vec{A} \).

Substituting for \( \vec{C} \):
\[ \vec{AC} = \left(\frac{\vec{A} + \vec{B}}{2}\right) - \vec{A} = \frac{\vec{B} - \vec{A}}{2} \]
Thus, we have:
\[ \vec{PA} + \frac{\vec{B} - \vec{A}}{2} = \vec{PC} \]

4. **In triangle \( PBC \)**:
Similarly, we can write:
\[ \vec{PB} + \vec{BC} = \vec{PC} \]
Where \( \vec{BC} = \vec{C} - \vec{B} \):
\[ \vec{BC} = \left(\frac{\vec{A} + \vec{B}}{2}\right) - \vec{B} = \frac{\vec{A} - \vec{B}}{2} \]
Thus:
\[ \vec{PB} + \frac{\vec{A} - \vec{B}}{2} = \vec{PC} \]

5. **Combining the Equations**:
Now, we can combine both equations:
\[ \vec{PA} + \frac{\vec{B} - \vec{A}}{2} + \vec{PB} + \frac{\vec{A} - \vec{B}}{2} = 2\vec{PC} \]
Simplifying this, we find:
\[ \vec{PA} + \vec{PB} + \frac{\vec{B} - \vec{A} + \vec{A} - \vec{B}}{2} = 2\vec{PC} \]
The terms \( \vec{B} - \vec{A} \) and \( \vec{A} - \vec{B} \) cancel each other out, leading to:
\[ \vec{PA} + \vec{PB} = 2\vec{PC} \]

### Conclusion:
Thus, the correct relationship is:
\[ \vec{PA} + \vec{PB} = 2\vec{PC} \]
The answer is option (D).