(i) If `vecOA = a, vecOB = b`, then the vector area of triangle OAB is ........and the vector area of triangle ABC is .........where `vecOC= c`
(ii) If a, b, c are vectors from origin to the point A,B, C then`(a xx b + b xx c + c xx a)` is ........... to plane ABC.
(iii) Vertices of a triangle are `(1,2, 4), (3, 1, -2)` and `(4, 3, 1)` then its area is .......

Answer

Let's solve the given question step by step.

### Part (i)

1. **Vector Area of Triangle OAB**:
- The area of triangle OAB can be calculated using the formula:
\[ \text{Area} = \frac{1}{2} \vec{OA} \times \vec{OB} \]
- Given that \( \vec{OA} = \vec{a} \) and \( \vec{OB} = \vec{b} \), we can substitute these into the formula:
\[ \text{Area of triangle OAB} = \frac{1}{2} \vec{a} \times \vec{b} \]

2. **Vector Area of Triangle ABC**:
- The area of triangle ABC can be calculated using the formula:
\[ \text{Area} = \frac{1}{2} \left( \vec{AB} \times \vec{AC} \right) \]
- Here, \( \vec{AB} = \vec{b} - \vec{a} \) and \( \vec{AC} = \vec{c} - \vec{a} \). Thus, we can express the area as:
\[ \text{Area of triangle ABC} = \frac{1}{2} \left( \vec{b} \times \vec{c} + \vec{c} \times \vec{a} + \vec{a} \times \vec{b} \right) \]

### Part (ii)

- The expression \( \vec{a} \times \vec{b} + \vec{b} \times \vec{c} + \vec{c} \times \vec{a} \) is a vector that is perpendicular to the plane formed by the vectors \( \vec{a}, \vec{b}, \vec{c} \). Therefore, we conclude:
\[ \vec{a} \times \vec{b} + \vec{b} \times \vec{c} + \vec{c} \times \vec{a} \text{ is perpendicular to plane ABC.} \]

### Part (iii)

1. **Vertices of the Triangle**:
- The vertices of the triangle are given as \( A(1, 2, 4) \), \( B(3, 1, -2) \), and \( C(4, 3, 1) \).

2. **Vectors from Origin**:
- We can denote these points as vectors:
\[ \vec{A} = \begin{pmatrix} 1 \\ 2 \\ 4 \end{pmatrix}, \quad \vec{B} = \begin{pmatrix} 3 \\ 1 \\ -2 \end{pmatrix}, \quad \vec{C} = \begin{pmatrix} 4 \\ 3 \\ 1 \end{pmatrix} \]

3. **Calculating Cross Products**:
- First, calculate \( \vec{AB} = \vec{B} - \vec{A} \):
\[ \vec{AB} = \begin{pmatrix} 3 - 1 \\ 1 - 2 \\ -2 - 4 \end{pmatrix} = \begin{pmatrix} 2 \\ -1 \\ -6 \end{pmatrix} \]
- Next, calculate \( \vec{AC} = \vec{C} - \vec{A} \):
\[ \vec{AC} = \begin{pmatrix} 4 - 1 \\ 3 - 2 \\ 1 - 4 \end{pmatrix} = \begin{pmatrix} 3 \\ 1 \\ -3 \end{pmatrix} \]

4. **Area Calculation**:
- The area of triangle ABC is given by:
\[ \text{Area} = \frac{1}{2} \left| \vec{AB} \times \vec{AC} \right| \]
- Calculate the cross product \( \vec{AB} \times \vec{AC} \):
\[ \vec{AB} \times \vec{AC} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ 2 & -1 & -6 \\ 3 & 1 & -3 \end{vmatrix} \]
- This determinant can be calculated to find the area.

5. **Final Area Calculation**:
- After calculating the determinant, we find:
\[ \text{Area} = \frac{1}{2} \sqrt{250} = \frac{5\sqrt{10}}{2} \text{ square units.} \]