Let `hata and hatb` be unit vectors at an angle `(pi)/(3)` with each other. If `(hatatimes(hatbtimeshatc))*(hatatimeshatc)=5` then
Statement-I `[hata hatb hatc]=10`
Statement-II [x y z]=0, if x=y or y=z or z=x

Options

A. Both Statement-I and Statement-II are correct and Statement-II is the correct explanation of Statement-I
B. Both Statement-I and Statement-II are correct but Statement-II is not the correct explanation of Statement-I
C. Statement-I is correct but Statement-II is incorrect
D. Statement-II is correct but Statement-I is incorrect

Answer

To solve the given problem step-by-step, we will analyze the statements and the given conditions.

### Given:
- Let \(\hat{a}\) and \(\hat{b}\) be unit vectors at an angle \(\frac{\pi}{3}\) with each other.
- We have the equation \((\hat{a} \times (\hat{b} \times \hat{c})) \cdot (\hat{a} \times \hat{c}) = 5\).

### Step 1: Understanding the Vector Triple Product
Using the vector triple product identity:
\[ \hat{a} \times (\hat{b} \times \hat{c}) = (\hat{a} \cdot \hat{c}) \hat{b} - (\hat{a} \cdot \hat{b}) \hat{c} \]
We can substitute this into our equation:
\[ (\hat{a} \cdot \hat{c}) \hat{b} - (\hat{a} \cdot \hat{b}) \hat{c} \]

### Step 2: Substitute into the Given Equation
Now, substituting this into the equation we have:
\[ ((\hat{a} \cdot \hat{c}) \hat{b} - (\hat{a} \cdot \hat{b}) \hat{c}) \cdot (\hat{a} \times \hat{c}) = 5 \]

### Step 3: Expand the Dot Product
Expanding the dot product:
\[ (\hat{a} \cdot \hat{c}) (\hat{b} \cdot (\hat{a} \times \hat{c})) - (\hat{a} \cdot \hat{b}) (\hat{c} \cdot (\hat{a} \times \hat{c})) = 5 \]

### Step 4: Evaluate the Terms
1. The term \(\hat{c} \cdot (\hat{a} \times \hat{c}) = 0\) because the dot product of a vector with itself is zero.
2. Therefore, the equation simplifies to:
\[ (\hat{a} \cdot \hat{c}) (\hat{b} \cdot (\hat{a} \times \hat{c})) = 5 \]

### Step 5: Find \(\hat{b} \cdot (\hat{a} \times \hat{c})\)
The term \(\hat{b} \cdot (\hat{a} \times \hat{c})\) represents the volume of the parallelepiped formed by the vectors \(\hat{a}, \hat{b}, \hat{c}\). This can be expressed as:
\[ [\hat{a} \hat{b} \hat{c}] = \hat{a} \cdot (\hat{b} \times \hat{c}) \]

### Step 6: Relate to the Given Condition
From the earlier steps, we can relate:
\[ (\hat{a} \cdot \hat{c}) [\hat{a} \hat{b} \hat{c}] = 5 \]
Let \(x = \hat{a} \cdot \hat{c}\) and \(y = [\hat{a} \hat{b} \hat{c}]\):
\[ x \cdot y = 5 \]

### Step 7: Evaluate \(\hat{a} \cdot \hat{c}\)
Since \(\hat{a}\) and \(\hat{b}\) are unit vectors at an angle \(\frac{\pi}{3}\), we have:
\[ \hat{a} \cdot \hat{b} = \cos\left(\frac{\pi}{3}\right) = \frac{1}{2} \]

### Step 8: Find the Magnitude of \([\hat{a} \hat{b} \hat{c}]\)
Using the relationship:
\[ [\hat{a} \hat{b} \hat{c}] = 10 \]
This implies that:
\[ x \cdot 10 = 5 \implies x = \frac{1}{2} \]

### Conclusion
Both statements are true:
- Statement I: \([\hat{a} \hat{b} \hat{c}] = 10\)
- Statement II: \([x y z] = 0\) if \(x = y\) or \(y = z\) or \(z = x\)

Thus, the answer is that both statements are true.

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