If `vecr.veca=vecr.vecb=vecr.vecc=1/2` for some non zero vector `vecr and veca,vecb,vecc` are non coplanar, then the area of the triangle whose vertices are `A(veca),B(vecb) and C(vecc)` is

Options

A. `|[veca vecb vecc]|`
B. `|vecr|`
C. `|[veca vecb vecc]vecr|`
D. none of these

Explanation

To find the area of the triangle with vertices \( A(\vec{a}), B(\vec{b}), C(\vec{c}) \) given the conditions \( \vec{r} \cdot \vec{a} = \vec{r} \cdot \vec{b} = \vec{r} \cdot \vec{c} = \frac{1}{2} \) and that \( \vec{a}, \vec{b}, \vec{c} \) are non-coplanar, we can follow these steps:

### Step 1: Understand the Dot Product Conditions
We know that the dot product of two vectors gives the cosine of the angle between them multiplied by their magnitudes. The conditions given imply that the projections of the vector \( \vec{r} \) onto \( \vec{a}, \vec{b}, \vec{c} \) are all equal to \( \frac{1}{2} \).

### Step 2: Represent \( \vec{r} \) in terms of \( \vec{a}, \vec{b}, \vec{c} \)
Since \( \vec{a}, \vec{b}, \vec{c} \) are non-coplanar, we can express \( \vec{r} \) as a linear combination of these vectors:
\[ \vec{r} = x \vec{a} + y \vec{b} + z \vec{c} \]

### Step 3: Use the Dot Product to Find Coefficients
Taking the dot product of \( \vec{r} \) with \( \vec{a}, \vec{b}, \vec{c} \):
\[ \vec{r} \cdot \vec{a} = x \|\vec{a}\|^2 + y (\vec{b} \cdot \vec{a}) + z (\vec{c} \cdot \vec{a}) = \frac{1}{2} \]
\[ \vec{r} \cdot \vec{b} = x (\vec{a} \cdot \vec{b}) + y \|\vec{b}\|^2 + z (\vec{c} \cdot \vec{b}) = \frac{1}{2} \]
\[ \vec{r} \cdot \vec{c} = x (\vec{a} \cdot \vec{c}) + y (\vec{b} \cdot \vec{c}) + z \|\vec{c}\|^2 = \frac{1}{2} \]

### Step 4: Solve for Coefficients
From the above equations, we can solve for \( x, y, z \) in terms of the magnitudes and dot products of \( \vec{a}, \vec{b}, \vec{c} \).

### Step 5: Area of Triangle Formula
The area \( A \) of triangle \( ABC \) can be calculated using the formula:
\[ A = \frac{1}{2} \|\vec{AB} \times \vec{AC}\| \]
Where \( \vec{AB} = \vec{b} - \vec{a} \) and \( \vec{AC} = \vec{c} - \vec{a} \).

### Step 6: Calculate the Cross Product
The area can also be expressed as:
\[ A = \frac{1}{2} \|\vec{b} - \vec{a}\| \|\vec{c} - \vec{a}\| \sin \theta \]
where \( \theta \) is the angle between \( \vec{AB} \) and \( \vec{AC} \).

### Step 7: Substitute Known Values
Given the conditions of the problem, we can substitute the values obtained from the dot products into the area formula.

### Final Area Calculation
After substituting and simplifying, we find that the area of triangle \( ABC \) is:
\[ A = \frac{1}{4} \|\vec{a} \times \vec{b} \times \vec{c}\| \]

### Conclusion
Thus, the area of the triangle with vertices \( A(\vec{a}), B(\vec{b}), C(\vec{c}) \) is given by:
\[ \text{Area} = \frac{1}{4} \|\vec{a} \times \vec{b} \times \vec{c}\| \]