If `(vecaxxvecb)xxvecc=vecaxx(vecbxxvecc)` where `veca, vecb` and `vecc` are any three vectors such that `veca.vecb!=0, vecb. vecc!=0` then `veca` and `vecc` are

Options

A. inclined at angle `(pi)/3` between them
B. inclined at angle of `(pi)/6` between them
C. perpendicular
D. parallel

Explanation

To solve the problem, we need to analyze the given equation:

\[ (\vec{a} \times \vec{b}) \times \vec{c} = \vec{a} \times (\vec{b} \times \vec{c}) \]

where \(\vec{a}, \vec{b}, \vec{c}\) are vectors such that \(\vec{a} \cdot \vec{b} \neq 0\) and \(\vec{b} \cdot \vec{c} \neq 0\).

### Step 1: Use the Vector Triple Product Identity
We will apply the vector triple product identity, which states that:

\[ \vec{x} \times (\vec{y} \times \vec{z}) = (\vec{x} \cdot \vec{z}) \vec{y} - (\vec{x} \cdot \vec{y}) \vec{z} \]

Using this identity, we can rewrite both sides of the equation.

### Step 2: Rewrite the Left Side
For the left side, \((\vec{a} \times \vec{b}) \times \vec{c}\):

\[ (\vec{a} \times \vec{b}) \times \vec{c} = (\vec{a} \cdot \vec{c}) \vec{b} - (\vec{b} \cdot \vec{c}) \vec{a} \]

### Step 3: Rewrite the Right Side
For the right side, \(\vec{a} \times (\vec{b} \times \vec{c})\):

\[ \vec{a} \times (\vec{b} \times \vec{c}) = (\vec{a} \cdot \vec{c}) \vec{b} - (\vec{a} \cdot \vec{b}) \vec{c} \]

### Step 4: Set the Two Sides Equal
Now we set the two expressions equal to each other:

\[ (\vec{a} \cdot \vec{c}) \vec{b} - (\vec{b} \cdot \vec{c}) \vec{a} = (\vec{a} \cdot \vec{c}) \vec{b} - (\vec{a} \cdot \vec{b}) \vec{c} \]

### Step 5: Simplify the Equation
By simplifying the equation, we can cancel out the common terms:

\[ -(\vec{b} \cdot \vec{c}) \vec{a} = -(\vec{a} \cdot \vec{b}) \vec{c} \]

This leads to:

\[ (\vec{b} \cdot \vec{c}) \vec{a} = (\vec{a} \cdot \vec{b}) \vec{c} \]

### Step 6: Rearranging the Equation
Rearranging gives us:

\[ \vec{a} = \frac{\vec{a} \cdot \vec{b}}{\vec{b} \cdot \vec{c}} \vec{c} \]

### Step 7: Conclusion
Since \(\frac{\vec{a} \cdot \vec{b}}{\vec{b} \cdot \vec{c}}\) is a scalar, we conclude that:

\[ \vec{a} \text{ is parallel to } \vec{c} \]

### Final Answer
Thus, the vectors \(\vec{a}\) and \(\vec{c}\) are parallel.

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