If ` veca , vecb, vecc` are the position vectors of the vertices. A,B,C of a triangle ABC. Then the area of triangle ABC is

Options

A. `vecaxxvecb+vecbxxvecc+veccxxveca`
B. `(1)/(2)(vecaxxvecb).vecc`
C. `(1)/(2)|vecaxxvecb|`
D. `(1)/(2)|vecaxxvecb+vecbxxvecc+veccxxveca|`

Answer

To find the area of triangle ABC given the position vectors of its vertices A, B, and C, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Position Vectors:**
Let the position vectors of the vertices A, B, and C be represented as:
\[ \vec{A}, \vec{B}, \vec{C} \]

2. **Determine the Vectors for Two Sides:**
To find the area of triangle ABC, we need the vectors representing two adjacent sides of the triangle. We can choose:
\[ \vec{AB} = \vec{B} - \vec{A} \]
\[ \vec{AC} = \vec{C} - \vec{A} \]

3. **Use the Cross Product to Find the Area:**
The area \(A\) of triangle ABC can be calculated using the formula:
\[ \text{Area} = \frac{1}{2} |\vec{AB} \times \vec{AC}| \]
Substituting the expressions for \(\vec{AB}\) and \(\vec{AC}\):
\[ \text{Area} = \frac{1}{2} |(\vec{B} - \vec{A}) \times (\vec{C} - \vec{A})| \]

4. **Expand the Cross Product:**
Using the properties of the cross product, we can expand this:
\[ \vec{AB} \times \vec{AC} = (\vec{B} - \vec{A}) \times (\vec{C} - \vec{A}) = \vec{B} \times \vec{C} - \vec{B} \times \vec{A} - \vec{A} \times \vec{C} + \vec{A} \times \vec{A} \]
Since \(\vec{A} \times \vec{A} = \vec{0}\), we simplify to:
\[ \vec{AB} \times \vec{AC} = \vec{B} \times \vec{C} - \vec{B} \times \vec{A} - \vec{A} \times \vec{C} \]

5. **Final Area Expression:**
Thus, the area of triangle ABC can be expressed as:
\[ \text{Area} = \frac{1}{2} |\vec{B} \times \vec{C} + \vec{A} \times \vec{B} + \vec{C} \times \vec{A}| \]

### Final Result:
The area of triangle ABC is given by:
\[ \text{Area} = \frac{1}{2} |\vec{B} \times \vec{C} + \vec{A} \times \vec{B} + \vec{C} \times \vec{A}| \]