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 determine if the expression \((\mathbf{a} \times \mathbf{b}) \times (\mathbf{c} \times \mathbf{d})\) results in a vector. We will analyze this step by step.

### Step 1: Understand 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 both \(\mathbf{u}\) and \(\mathbf{v}\). Therefore, the result of a cross product is always a vector.

**Hint:** Remember that the cross product of two vectors always yields another vector that is orthogonal to the plane formed by the two original vectors.

### Step 2: Analyze the Expression
We have the expression \((\mathbf{a} \times \mathbf{b}) \times (\mathbf{c} \times \mathbf{d})\). Let's denote:
- \(\mathbf{m} = \mathbf{a} \times \mathbf{b}\)
- \(\mathbf{n} = \mathbf{c} \times \mathbf{d}\)

Now, we can rewrite the expression as \(\mathbf{m} \times \mathbf{n}\).

**Hint:** Identify the intermediate vectors formed by the cross products to simplify the expression.

### Step 3: Determine the Result of \(\mathbf{m} \times \mathbf{n}\)
Since both \(\mathbf{m}\) and \(\mathbf{n}\) are vectors (as established in Step 1), the cross product \(\mathbf{m} \times \mathbf{n}\) will also yield a vector.

**Hint:** The cross product of any two vectors results in another vector, which is crucial for confirming the overall expression.

### Step 4: Conclusion
Thus, the expression \((\mathbf{a} \times \mathbf{b}) \times (\mathbf{c} \times \mathbf{d})\) is indeed a vector.

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