If `hata, hatb, hatc` are three units vectors such that `hatb` and `hatc` are non-parallel and `hataxx(hatbxxhatc)=1//2hatb` then the angle between `hata` and `hatc` is

Options

A. `30^(@)`
B. `45^(@)`
C. `60^(@)`
D. `90^(@)`

Explanation

To solve the problem step by step, we need to analyze the given information and apply vector properties accordingly.

### Step 1: Understand the given vectors
We have three unit vectors: \( \hat{a}, \hat{b}, \hat{c} \). The vectors \( \hat{b} \) and \( \hat{c} \) are non-parallel.

### Step 2: Write down the given equation
The problem states that:
\[ \hat{a} \times (\hat{b} \times \hat{c}) = \frac{1}{2} \hat{b} \]

### Step 3: Use the vector triple product identity
We can use the vector triple product identity:
\[ \hat{x} \times (\hat{y} \times \hat{z}) = (\hat{x} \cdot \hat{z}) \hat{y} - (\hat{x} \cdot \hat{y}) \hat{z} \]
Applying this to our case:
\[ \hat{a} \times (\hat{b} \times \hat{c}) = (\hat{a} \cdot \hat{c}) \hat{b} - (\hat{a} \cdot \hat{b}) \hat{c} \]

### Step 4: Set the equation equal to the given expression
From the problem, we have:
\[ (\hat{a} \cdot \hat{c}) \hat{b} - (\hat{a} \cdot \hat{b}) \hat{c} = \frac{1}{2} \hat{b} \]

### Step 5: Compare coefficients
Since \( \hat{b} \) and \( \hat{c} \) are non-parallel, we can equate the coefficients of \( \hat{b} \) and \( \hat{c} \) separately.

1. Coefficient of \( \hat{b} \):
\[ \hat{a} \cdot \hat{c} = \frac{1}{2} \]

2. Coefficient of \( \hat{c} \):
\[ -(\hat{a} \cdot \hat{b}) = 0 \implies \hat{a} \cdot \hat{b} = 0 \]

### Step 6: Interpret the results
From \( \hat{a} \cdot \hat{b} = 0 \), we conclude that \( \hat{a} \) is perpendicular to \( \hat{b} \).

From \( \hat{a} \cdot \hat{c} = \frac{1}{2} \), we can express this in terms of the angle \( \alpha \) between \( \hat{a} \) and \( \hat{c} \):
\[ \hat{a} \cdot \hat{c} = |\hat{a}| |\hat{c}| \cos(\alpha) \]
Since both \( \hat{a} \) and \( \hat{c} \) are unit vectors, this simplifies to:
\[ \cos(\alpha) = \frac{1}{2} \]

### Step 7: Find the angle
The angle \( \alpha \) for which \( \cos(\alpha) = \frac{1}{2} \) is:
\[ \alpha = 60^\circ \]

### Conclusion
Thus, the angle between \( \hat{a} \) and \( \hat{c} \) is \( 60^\circ \).

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