IF `bara,barb,barc and bard` are any four vectors then `(bara times barb) times (barc times bard)` is a vector

Options

A. perpendicular to `bara,barb,barc and bard`
B. along the line of intersection of two planes, one containing `bara,barb` other containing `barc,bard`
C. equally inclined to both `bara times barb and barc times bard`
D. None of these

Answer

To solve the problem, we need to analyze the expression \((\mathbf{a} \times \mathbf{b}) \times (\mathbf{c} \times \mathbf{d})\) and determine whether it results in a vector.

### Step-by-Step Solution:

1. **Understanding the Cross Product**:
The cross product of two vectors \(\mathbf{u}\) and \(\mathbf{v}\), denoted as \(\mathbf{u} \times \mathbf{v}\), results in a vector that is perpendicular to the plane formed by \(\mathbf{u}\) and \(\mathbf{v}\).

2. **First Cross Product**:
Let's denote \(\mathbf{m} = \mathbf{a} \times \mathbf{b}\).
- Here, \(\mathbf{m}\) is a vector that is perpendicular to both \(\mathbf{a}\) and \(\mathbf{b}\).

3. **Second Cross Product**:
Similarly, let \(\mathbf{n} = \mathbf{c} \times \mathbf{d}\).
- The vector \(\mathbf{n}\) is perpendicular to both \(\mathbf{c}\) and \(\mathbf{d}\).

4. **Cross Product of Two Vectors**:
Now, we need to evaluate \(\mathbf{m} \times \mathbf{n}\).
- The result of this cross product will be a vector that is perpendicular to the plane formed by \(\mathbf{m}\) and \(\mathbf{n}\).

5. **Properties of Cross Product**:
The cross product \(\mathbf{m} \times \mathbf{n}\) is defined and will yield a vector. Therefore, \((\mathbf{a} \times \mathbf{b}) \times (\mathbf{c} \times \mathbf{d})\) is indeed a vector.

6. **Conclusion**:
Since both \(\mathbf{m}\) and \(\mathbf{n}\) are vectors, and the cross product of two vectors results in another vector, we conclude that \((\mathbf{a} \times \mathbf{b}) \times (\mathbf{c} \times \mathbf{d})\) is a vector.

### Final Answer:
Yes, \((\mathbf{a} \times \mathbf{b}) \times (\mathbf{c} \times \mathbf{d})\) is a vector.

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