The position vectors of the vertices A, B and C of a triangle are three unit vectors `veca, vecb and vecc` respectively. A vector `vecd` is such that `vecd.hata =vecd. Hatb=vecd.hatc and vecd= lambda (hatb + hatc)` . Then triangle ABC is

Options

A. acute angled
B. obtuse angled
C. right angled
D. none of these

Explanation

To solve the problem, we need to analyze the given conditions about the triangle ABC and the vector \( \vec{d} \).

### Step-by-Step Solution:

1. **Understanding the Given Vectors**:
- The position vectors of the vertices A, B, and C are given as unit vectors \( \vec{a}, \vec{b}, \vec{c} \).
- We also know that \( \vec{d} \) satisfies the condition \( \vec{d} \cdot \hat{a} = \vec{d} \cdot \hat{b} = \vec{d} \cdot \hat{c} \).

2. **Substituting the Expression for \( \vec{d} \)**:
- We have \( \vec{d} = \lambda (\hat{b} + \hat{c}) \).
- Substituting this into the dot product conditions gives us:
\[ \lambda (\hat{b} + \hat{c}) \cdot \hat{a} = \lambda (\hat{b} + \hat{c}) \cdot \hat{b} = \lambda (\hat{b} + \hat{c}) \cdot \hat{c} \]

3. **Expanding the Dot Products**:
- This leads to:
\[ \lambda (\hat{b} \cdot \hat{a} + \hat{c} \cdot \hat{a}) = \lambda (1 + \hat{c} \cdot \hat{b}) = \lambda (\hat{b} \cdot \hat{c} + 1) \]
- Simplifying gives:
\[ \hat{b} \cdot \hat{a} + \hat{c} \cdot \hat{a} = 1 + \hat{b} \cdot \hat{c} \]

4. **Rearranging the Equation**:
- Rearranging the equation leads to:
\[ \hat{b} \cdot \hat{a} + \hat{c} \cdot \hat{a} - \hat{b} \cdot \hat{c} - 1 = 0 \]

5. **Interpreting the Result**:
- This equation can be interpreted geometrically. If we let \( \hat{a} - \hat{c} \) and \( \hat{a} - \hat{b} \) represent the sides of the triangle, we can analyze their relationships.
- Specifically, if we can show that the vectors \( \hat{a} - \hat{c} \) and \( \hat{a} - \hat{b} \) are perpendicular, then triangle ABC is a right triangle.

6. **Final Conclusion**:
- From the derived equation, we can conclude that the triangle ABC is a right-angled triangle, with the right angle at vertex A.