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

Options

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

Answer

To solve the problem, we need to analyze the given vector equation and the conditions provided. The equation we have is:

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

We also know the following conditions:

1. \(\vec{a} \cdot \vec{b} = 0\) (which means \(\vec{a}\) is perpendicular to \(\vec{b}\))
2. \(\vec{b} \cdot \vec{c} = 0\) (which means \(\vec{b}\) is perpendicular to \(\vec{c}\))

### Step 1: Apply the Vector Triple Product Identity

Using the vector triple product identity, we can rewrite both sides of the equation:

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

Applying this to both sides:

- Left-hand side:
\[ (\vec{a} \times \vec{b}) \times \vec{c} = (\vec{a} \cdot \vec{c}) \vec{b} - (\vec{b} \cdot \vec{c}) \vec{a} \]

- Right-hand side:
\[ \vec{a} \times (\vec{b} \times \vec{c}) = (\vec{b} \cdot \vec{c}) \vec{a} - (\vec{a} \cdot \vec{b}) \vec{c} \]

### Step 2: Set the Two Sides Equal

Now, we set the two results equal to each other:

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

### Step 3: Simplify the Equation

Rearranging the equation gives us:

\[ (\vec{a} \cdot \vec{c}) \vec{b} = 2(\vec{b} \cdot \vec{c}) \vec{a} + (\vec{a} \cdot \vec{b}) \vec{c} \]

### Step 4: Analyze the Conditions

Since \(\vec{a} \cdot \vec{b} = 0\) and \(\vec{b} \cdot \vec{c} = 0\), we can substitute these values into our equation:

\[ (\vec{a} \cdot \vec{c}) \vec{b} = 0 \]

This implies that either \(\vec{b} = 0\) (which is not the case as it is a vector) or \(\vec{a} \cdot \vec{c} = 0\). Therefore, \(\vec{a}\) is perpendicular to \(\vec{c}\).

### Step 5: Conclusion

Since we have established that \(\vec{a} \cdot \vec{c} = 0\), we conclude that \(\vec{a}\) and \(\vec{c}\) are perpendicular to each other.

### Final Answer

Thus, the answer to the question is that \(\vec{a}\) and \(\vec{c}\) are **perpendicular**.

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