If `vec(a), vec(b), vec(c )` are the position vectors of the vecrtices A, B, C of a `Delta ABC` respectively, find an expression for the area of `Delta ABC` and hence deduce the condition for the points A, B, C to be collinear.

Answer

To find the area of triangle ABC given the position vectors \(\vec{a}\), \(\vec{b}\), and \(\vec{c}\) of vertices A, B, and C respectively, we can follow these steps:

### Step 1: Define the vectors
Let:
- \(\vec{A} = \vec{a}\) (position vector of vertex A)
- \(\vec{B} = \vec{b}\) (position vector of vertex B)
- \(\vec{C} = \vec{c}\) (position vector of vertex C)

### Step 2: Find the vectors representing the sides of the triangle
The vectors representing the sides of triangle ABC can be defined as:
- \(\vec{AB} = \vec{B} - \vec{A} = \vec{b} - \vec{a}\)
- \(\vec{AC} = \vec{C} - \vec{A} = \vec{c} - \vec{a}\)

### Step 3: Use the cross product to find the area
The area \(A\) of triangle ABC can be calculated using the formula:
\[ A = \frac{1}{2} \left| \vec{AB} \times \vec{AC} \right| \]
Substituting the expressions for \(\vec{AB}\) and \(\vec{AC}\):
\[ A = \frac{1}{2} \left| (\vec{b} - \vec{a}) \times (\vec{c} - \vec{a}) \right| \]

### Step 4: Simplify the expression
Using the properties of the cross product:
\[ A = \frac{1}{2} \left| \vec{b} - \vec{a} \times \vec{c} - \vec{a} \right| \]
This can be rewritten as:
\[ A = \frac{1}{2} \left| (\vec{c} - \vec{a}) \times (\vec{b} - \vec{a}) \right| \]

### Step 5: Condition for collinearity
For points A, B, and C to be collinear, the area of triangle ABC must be zero:
\[ \frac{1}{2} \left| (\vec{c} - \vec{a}) \times (\vec{b} - \vec{a}) \right| = 0 \]
This implies:
\[ \left| (\vec{c} - \vec{a}) \times (\vec{b} - \vec{a}) \right| = 0 \]
The cross product is zero when the vectors are parallel, which indicates that points A, B, and C are collinear.

### Final Expression for Area
Thus, the area of triangle ABC is given by:
\[ A = \frac{1}{2} \left| (\vec{b} - \vec{a}) \times (\vec{c} - \vec{a}) \right| \]
And the condition for collinearity of points A, B, and C is:
\[ (\vec{c} - \vec{a}) \times (\vec{b} - \vec{a}) = \vec{0} \]