Let `veca, vecb and vecc ` be three non zero vectors such that `vec b.vec c= 0` and `veca times(vecb xx vecc)=(vecb-vecc)/2`. If `vecd` be a vector such that `vecb.vecd = veca .vecb `, then `(veca times vecb).(vecc times vecd)` is equal to

Options

A. `3/4`
B. `1/2`
C. `-1/4`
D. `1/4`

Answer

To solve the problem, we need to analyze the given vectors and their relationships step by step.

### Step 1: Understand the given conditions
We have three non-zero vectors \( \vec{a}, \vec{b}, \vec{c} \) such that:
1. \( \vec{b} \cdot \vec{c} = 0 \) (which means \( \vec{b} \) and \( \vec{c} \) are orthogonal)
2. \( \vec{a} \times (\vec{b} \times \vec{c}) = \frac{\vec{b} - \vec{c}}{2} \)

### Step 2: Use the vector triple product identity
We can use the vector triple product identity:
\[ \vec{u} \times (\vec{v} \times \vec{w}) = (\vec{u} \cdot \vec{w}) \vec{v} - (\vec{u} \cdot \vec{v}) \vec{w} \]
Applying this to \( \vec{a} \times (\vec{b} \times \vec{c}) \):
\[ \vec{a} \times (\vec{b} \times \vec{c}) = (\vec{a} \cdot \vec{c}) \vec{b} - (\vec{a} \cdot \vec{b}) \vec{c} \]
Setting this equal to \( \frac{\vec{b} - \vec{c}}{2} \):
\[ (\vec{a} \cdot \vec{c}) \vec{b} - (\vec{a} \cdot \vec{b}) \vec{c} = \frac{\vec{b}}{2} - \frac{\vec{c}}{2} \]

### Step 3: Equate coefficients
From the equation, we can equate coefficients of \( \vec{b} \) and \( \vec{c} \):
1. \( \vec{a} \cdot \vec{c} = \frac{1}{2} \)
2. \( -(\vec{a} \cdot \vec{b}) = -\frac{1}{2} \) which gives \( \vec{a} \cdot \vec{b} = \frac{1}{2} \)

### Step 4: Analyze vector \( \vec{d} \)
We are given that \( \vec{b} \cdot \vec{d} = \vec{a} \cdot \vec{b} \). Since \( \vec{a} \cdot \vec{b} = \frac{1}{2} \), we have:
\[ \vec{b} \cdot \vec{d} = \frac{1}{2} \]

### Step 5: Calculate \( (\vec{a} \times \vec{b}) \cdot (\vec{c} \times \vec{d}) \)
Using the identity for the dot product of cross products:
\[ \vec{u} \cdot (\vec{v} \times \vec{w}) = \vec{v} \cdot (\vec{u} \times \vec{w}) \]
We can write:
\[ (\vec{a} \times \vec{b}) \cdot (\vec{c} \times \vec{d}) = (\vec{c} \cdot \vec{a}) (\vec{b} \cdot \vec{d}) - (\vec{b} \cdot \vec{a}) (\vec{c} \cdot \vec{d}) \]

### Step 6: Substitute known values
Substituting the known values:
- \( \vec{c} \cdot \vec{a} = \frac{1}{2} \)
- \( \vec{b} \cdot \vec{a} = \frac{1}{2} \)
- \( \vec{b} \cdot \vec{d} = \frac{1}{2} \)

Thus:
\[ (\vec{a} \times \vec{b}) \cdot (\vec{c} \times \vec{d}) = \left(\frac{1}{2}\right) \left(\frac{1}{2}\right) - \left(\frac{1}{2}\right) (\vec{c} \cdot \vec{d}) \]

### Step 7: Determine \( \vec{c} \cdot \vec{d} \)
Since \( \vec{b} \cdot \vec{d} = \frac{1}{2} \), we can find \( \vec{c} \cdot \vec{d} \) using the orthogonality condition and the relationships derived.

### Final Calculation
Assuming \( \vec{c} \cdot \vec{d} = 0 \) (due to orthogonality), we have:
\[ (\vec{a} \times \vec{b}) \cdot (\vec{c} \times \vec{d}) = \frac{1}{4} - 0 = \frac{1}{4} \]

### Conclusion
Thus, the final answer is:
\[ \boxed{\frac{1}{4}} \]