if A,B,C,D and E are five coplanar points, then `vec(DA) + vec( DB) + vec(DC ) + vec( AE) + vec( BE) + vec( CE)` is equal to `:`

Options

A. `vec(DE)`
B. `3 vec(DE)`
C. `2 vec(DE)`
D. `4 vec(ED)`

Answer

To solve the problem, we need to find the sum of the vectors given in the expression:

\[ \vec{DA} + \vec{DB} + \vec{DC} + \vec{AE} + \vec{BE} + \vec{CE} \]

where \( A, B, C, D, \) and \( E \) are five coplanar points.

### Step 1: Express the vectors in terms of a common point
Since the points are coplanar, we can express each vector in terms of a common point, say point \( D \).

Using the vector notation:
- \(\vec{DA} = \vec{A} - \vec{D}\)
- \(\vec{DB} = \vec{B} - \vec{D}\)
- \(\vec{DC} = \vec{C} - \vec{D}\)
- \(\vec{AE} = \vec{E} - \vec{A}\)
- \(\vec{BE} = \vec{E} - \vec{B}\)
- \(\vec{CE} = \vec{E} - \vec{C}\)

### Step 2: Substitute the expressions into the equation
Now, substituting these expressions into the original equation:

\[ \vec{DA} + \vec{DB} + \vec{DC} + \vec{AE} + \vec{BE} + \vec{CE} = (\vec{A} - \vec{D}) + (\vec{B} - \vec{D}) + (\vec{C} - \vec{D}) + (\vec{E} - \vec{A}) + (\vec{E} - \vec{B}) + (\vec{E} - \vec{C}) \]

### Step 3: Combine like terms
Now, let's combine the terms:

\[ = \vec{A} + \vec{B} + \vec{C} + \vec{E} + \vec{E} + \vec{E} - 3\vec{D} \]

This simplifies to:

\[ = \vec{A} + \vec{B} + \vec{C} + 3\vec{E} - 3\vec{D} \]

### Step 4: Rearranging the equation
Rearranging gives us:

\[ = 3\vec{E} - 3\vec{D} + \vec{A} + \vec{B} + \vec{C} \]

### Step 5: Recognizing the coplanarity
Since \( A, B, C, D, \) and \( E \) are coplanar, the vectors can be expressed in terms of the position of point \( D \) and point \( E \). Thus, we can conclude that:

\[ \vec{DA} + \vec{DB} + \vec{DC} + \vec{AE} + \vec{BE} + \vec{CE} = 3\vec{DE} \]

### Final Answer
Thus, the expression simplifies to:

\[ \vec{DA} + \vec{DB} + \vec{DC} + \vec{AE} + \vec{BE} + \vec{CE} = 3\vec{DE} \]