Let `vec a, vec b, vec c` are three vectors such that `veca .veca=vecb . vecb = vecc . vecc = 3 and |veca-vecb|^2+|vecb-vecc|^2+|vecc-veca|^2=27,` then

Options

A. a,b and c are necessarily coplanar
B. a,b and c represent sides of a triangle in magnitude and direction
C. `a*b+b*c+c*a` has the least value `-9//2`
D. a,b and c represent orthogonal triad of vectors

Answer

To solve the problem step-by-step, we start with the given information about the vectors \(\vec{a}\), \(\vec{b}\), and \(\vec{c}\):

1. **Given Conditions**:
- \(\vec{a} \cdot \vec{a} = \vec{b} \cdot \vec{b} = \vec{c} \cdot \vec{c} = 3\)
- \(|\vec{a} - \vec{b}|^2 + |\vec{b} - \vec{c}|^2 + |\vec{c} - \vec{a}|^2 = 27\)

2. **Using the Dot Product**:
The expression \(|\vec{x} - \vec{y}|^2\) can be expanded using the dot product:
\[ |\vec{x} - \vec{y}|^2 = (\vec{x} - \vec{y}) \cdot (\vec{x} - \vec{y}) = \vec{x} \cdot \vec{x} - 2\vec{x} \cdot \vec{y} + \vec{y} \cdot \vec{y} \]

3. **Expanding Each Term**:
- For \(|\vec{a} - \vec{b}|^2\):
\[ |\vec{a} - \vec{b}|^2 = \vec{a} \cdot \vec{a} - 2\vec{a} \cdot \vec{b} + \vec{b} \cdot \vec{b} = 3 - 2\vec{a} \cdot \vec{b} + 3 = 6 - 2\vec{a} \cdot \vec{b} \]
- For \(|\vec{b} - \vec{c}|^2\):
\[ |\vec{b} - \vec{c}|^2 = \vec{b} \cdot \vec{b} - 2\vec{b} \cdot \vec{c} + \vec{c} \cdot \vec{c} = 3 - 2\vec{b} \cdot \vec{c} + 3 = 6 - 2\vec{b} \cdot \vec{c} \]
- For \(|\vec{c} - \vec{a}|^2\):
\[ |\vec{c} - \vec{a}|^2 = \vec{c} \cdot \vec{c} - 2\vec{c} \cdot \vec{a} + \vec{a} \cdot \vec{a} = 3 - 2\vec{c} \cdot \vec{a} + 3 = 6 - 2\vec{c} \cdot \vec{a} \]

4. **Combining the Expansions**:
Now, we combine all the expanded terms:
\[ (6 - 2\vec{a} \cdot \vec{b}) + (6 - 2\vec{b} \cdot \vec{c}) + (6 - 2\vec{c} \cdot \vec{a}) = 27 \]
Simplifying this gives:
\[ 18 - 2(\vec{a} \cdot \vec{b} + \vec{b} \cdot \vec{c} + \vec{c} \cdot \vec{a}) = 27 \]

5. **Solving for the Dot Products**:
Rearranging the equation:
\[ -2(\vec{a} \cdot \vec{b} + \vec{b} \cdot \vec{c} + \vec{c} \cdot \vec{a}) = 27 - 18 \]
\[ -2(\vec{a} \cdot \vec{b} + \vec{b} \cdot \vec{c} + \vec{c} \cdot \vec{a}) = 9 \]
Dividing by -2:
\[ \vec{a} \cdot \vec{b} + \vec{b} \cdot \vec{c} + \vec{c} \cdot \vec{a} = -\frac{9}{2} \]

6. **Conclusion**:
The least value of \(\vec{a} \cdot \vec{b} + \vec{b} \cdot \vec{c} + \vec{c} \cdot \vec{a}\) is \(-\frac{9}{2}\).