If `(veca xx vecb)xx vec c=veca xx (vecb xx vec c)`, where `veca, vecb and vec c` are any three vectors such that `veca*vecb ne 0, vecb*vec c ne 0`, then `veca and vec c` are :

Options

A. Inclined at an angle of `(pi)/(3)`
B. Inclined at an angle of `(pi)/(6)`
C. Perpendicular
D. Parallel

Answer

To solve the problem, we need to analyze the given vector 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 can 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, we can cancel \((\vec{a} \cdot \vec{c}) \vec{b}\) from both sides:

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

This leads to:

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

### Step 6: Analyze the Result
From the equation \((\vec{b} \cdot \vec{c}) \vec{a} = (\vec{a} \cdot \vec{b}) \vec{c}\), we can conclude that \(\vec{a}\) and \(\vec{c}\) are scalar multiples of each other, which implies that they are parallel.

### Conclusion
Thus, we can conclude that:

\(\vec{a}\) and \(\vec{c}\) are parallel vectors.