Prove that the resultant of the vectors represented by the sides `vec(AB)` and `vec(AC)` of a triangle ABC is `2 vec(AD)`, where D is the mid - point of [BC].

Answer

To prove that the resultant of the vectors represented by the sides \(\vec{AB}\) and \(\vec{AC}\) of triangle \(ABC\) is \(2 \vec{AD}\), where \(D\) is the midpoint of segment \(BC\), we can follow these steps:

### Step 1: Define the 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 \(C\),
- \(\vec{D}\) be the position vector of point \(D\), which is the midpoint of \(BC\).

Since \(D\) is the midpoint of \(BC\), we can express \(\vec{D}\) as:
\[ \vec{D} = \frac{\vec{B} + \vec{C}}{2} \]

### Step 2: Express \(\vec{AB}\) and \(\vec{AC}\)
The vectors \(\vec{AB}\) and \(\vec{AC}\) can be expressed as:
\[ \vec{AB} = \vec{B} - \vec{A} \]
\[ \vec{AC} = \vec{C} - \vec{A} \]

### Step 3: Find the Resultant of \(\vec{AB}\) and \(\vec{AC}\)
The resultant vector \(\vec{R}\) of \(\vec{AB}\) and \(\vec{AC}\) is given by:
\[ \vec{R} = \vec{AB} + \vec{AC} \]
Substituting the expressions for \(\vec{AB}\) and \(\vec{AC}\):
\[ \vec{R} = (\vec{B} - \vec{A}) + (\vec{C} - \vec{A}) = \vec{B} + \vec{C} - 2\vec{A} \]

### Step 4: Substitute \(\vec{D}\) into the Resultant
Now, we can substitute \(\vec{D}\) into our expression for \(\vec{R}\):
\[ \vec{R} = \vec{B} + \vec{C} - 2\vec{A} = 2\left(\frac{\vec{B} + \vec{C}}{2}\right) - 2\vec{A} = 2\vec{D} - 2\vec{A} \]

### Step 5: Rearranging the Resultant
We can rearrange this to express \(\vec{R}\) in terms of \(\vec{AD}\):
\[ \vec{R} = 2\left(\vec{D} - \vec{A}\right) = 2\vec{AD} \]

### Conclusion
Thus, we have shown that the resultant of the vectors represented by the sides \(\vec{AB}\) and \(\vec{AC}\) is:
\[ \vec{R} = 2\vec{AD} \]