Let the vectors `veca, vecb,vecc and vecd` be such that `(vecaxxvecb)xx(veccxxvecd)=vec0`. Let `P_1 and P_2` be planes determined by pairs of vectors `veca,vecb and vecc,vecd respectively. Then the angle between `P_1 and P_2` is (A) 0 (B) `pi/4` (C) `pi/3` (D) `pi/2`

Options

A. 0
B. `pi//4`
C. `pi//3`
D. `pi//2`

Explanation

To solve the problem, we will follow these steps:

### Step 1: Understand the given condition
We are given that \((\vec{a} \times \vec{b}) \times (\vec{c} \times \vec{d}) = \vec{0}\). This means that the vector resulting from the cross product of \(\vec{a} \times \vec{b}\) and \(\vec{c} \times \vec{d}\) is the zero vector.

### Step 2: Identify the planes
Let \(P_1\) be the plane determined by the vectors \(\vec{a}\) and \(\vec{b}\), and let \(P_2\) be the plane determined by the vectors \(\vec{c}\) and \(\vec{d}\).

### Step 3: Determine the normal vectors of the planes
The normal vector \(\vec{n_1}\) to the plane \(P_1\) can be found using the cross product:
\[ \vec{n_1} = \vec{a} \times \vec{b} \]
Similarly, the normal vector \(\vec{n_2}\) to the plane \(P_2\) is:
\[ \vec{n_2} = \vec{c} \times \vec{d} \]

### Step 4: Analyze the condition given
Since \((\vec{a} \times \vec{b}) \times (\vec{c} \times \vec{d}) = \vec{0}\), we can interpret this in terms of the normal vectors:
\[ \vec{n_1} \times \vec{n_2} = \vec{0} \]
This implies that \(\vec{n_1}\) and \(\vec{n_2}\) are parallel or one of them is the zero vector.

### Step 5: Conclude about the angle between the planes
If \(\vec{n_1}\) and \(\vec{n_2}\) are parallel, then the planes \(P_1\) and \(P_2\) are either the same or parallel to each other. This means that the angle \(\theta\) between the two planes is \(0\) degrees.

### Final Answer
Thus, the angle between the planes \(P_1\) and \(P_2\) is:
\[ \text{Angle} = 0 \text{ degrees} \]
The correct option is (A) 0.

---