In a `Delta ABC, D, E, F` are the mid -points of the sides BC, CA and AB respectively, the vector AD is equal to

Options

A. `vec(BE) + vec(C F)`
B. `vec(BE ) - vec(CF)`
C. `vec(CF) -vec(BE)`
D. `-vec(BE)- vec(CF)`

Answer

To find the vector \( \vec{AD} \) in triangle \( ABC \) where \( D, E, F \) are the midpoints of sides \( BC, CA, \) and \( AB \) respectively, we can follow these steps:

### Step 1: Define the Points
Let the position vectors of points \( A, B, \) and \( C \) be represented as \( \vec{A}, \vec{B}, \) and \( \vec{C} \).

### Step 2: Find the Midpoints
Using the midpoint formula, we can find the position vectors of the midpoints \( D, E, \) and \( F \):
- The midpoint \( D \) of side \( BC \) is given by:
\[ \vec{D} = \frac{\vec{B} + \vec{C}}{2} \]
- The midpoint \( E \) of side \( CA \) is given by:
\[ \vec{E} = \frac{\vec{C} + \vec{A}}{2} \]
- The midpoint \( F \) of side \( AB \) is given by:
\[ \vec{F} = \frac{\vec{A} + \vec{B}}{2} \]

### Step 3: Calculate the Vector \( \vec{AD} \)
The vector \( \vec{AD} \) can be calculated as:
\[ \vec{AD} = \vec{D} - \vec{A} \]
Substituting the expression for \( \vec{D} \):
\[ \vec{AD} = \left( \frac{\vec{B} + \vec{C}}{2} \right) - \vec{A} \]
This simplifies to:
\[ \vec{AD} = \frac{\vec{B} + \vec{C}}{2} - \vec{A} = \frac{\vec{B} + \vec{C} - 2\vec{A}}{2} \]

### Step 4: Final Expression
Thus, the vector \( \vec{AD} \) is given by:
\[ \vec{AD} = \frac{\vec{B} + \vec{C} - 2\vec{A}}{2} \]