If `veca, vecb, vecc, vecd` are on a circle of radius R whose centre is a origin and `vecc - veca` is perpendicular to `vecd.vecb`, then `|vecd-veca|^(2) +|vecb-vecc|^(2) = kR^(2)` where k is equal to………………

Answer

To solve the problem, we need to analyze the given vectors and their relationships based on the information provided. Let's break down the solution step by step.

### Step-by-Step Solution:

1. **Understanding the Vectors**:
Given that vectors \( \vec{a}, \vec{b}, \vec{c}, \vec{d} \) lie on a circle of radius \( R \) centered at the origin, we know:
\[ |\vec{a}| = |\vec{b}| = |\vec{c}| = |\vec{d}| = R \]

2. **Using the Perpendicular Condition**:
We are given that \( \vec{c} - \vec{a} \) is perpendicular to \( \vec{d} \cdot \vec{b} \). This means:
\[ (\vec{c} - \vec{a}) \cdot (\vec{d} - \vec{b}) = 0 \]

3. **Expanding the Expression**:
We need to find:
\[ |\vec{d} - \vec{a}|^2 + |\vec{b} - \vec{c}|^2 \]
This can be expanded using the dot product:
\[ |\vec{d} - \vec{a}|^2 = (\vec{d} - \vec{a}) \cdot (\vec{d} - \vec{a}) = |\vec{d}|^2 - 2\vec{d} \cdot \vec{a} + |\vec{a}|^2 \]
\[ |\vec{b} - \vec{c}|^2 = (\vec{b} - \vec{c}) \cdot (\vec{b} - \vec{c}) = |\vec{b}|^2 - 2\vec{b} \cdot \vec{c} + |\vec{c}|^2 \]

4. **Substituting Magnitudes**:
Since \( |\vec{a}| = |\vec{b}| = |\vec{c}| = |\vec{d}| = R \), we substitute:
\[ |\vec{d} - \vec{a}|^2 = R^2 - 2\vec{d} \cdot \vec{a} + R^2 = 2R^2 - 2\vec{d} \cdot \vec{a} \]
\[ |\vec{b} - \vec{c}|^2 = R^2 - 2\vec{b} \cdot \vec{c} + R^2 = 2R^2 - 2\vec{b} \cdot \vec{c} \]

5. **Combining the Results**:
Now, we combine the two expressions:
\[ |\vec{d} - \vec{a}|^2 + |\vec{b} - \vec{c}|^2 = (2R^2 - 2\vec{d} \cdot \vec{a}) + (2R^2 - 2\vec{b} \cdot \vec{c}) \]
\[ = 4R^2 - 2(\vec{d} \cdot \vec{a} + \vec{b} \cdot \vec{c}) \]

6. **Using the Perpendicular Condition**:
Since \( \vec{c} - \vec{a} \) is perpendicular to \( \vec{d} - \vec{b} \), we have:
\[ \vec{d} \cdot \vec{a} + \vec{b} \cdot \vec{c} = 0 \]
Thus:
\[ |\vec{d} - \vec{a}|^2 + |\vec{b} - \vec{c}|^2 = 4R^2 - 2(0) = 4R^2 \]

7. **Final Equation**:
We have:
\[ |\vec{d} - \vec{a}|^2 + |\vec{b} - \vec{c}|^2 = kR^2 \]
Comparing both sides gives:
\[ kR^2 = 4R^2 \implies k = 4 \]

### Conclusion:
The value of \( k \) is \( 4 \).