If `(veca xx vecb) xx (vecc xx vecd) . (veca xx vecd) =0` then which of the following may be true ?

Options

A. `veca, vecb ,vecc and vecd` are nenessarily coplanar
B. `veca` lies in the plane of `vecc and vecd`
C. `vecb` lies in the plane of `veca and vecd`
D. `vecc` lies in the plane of `veca and vecd`

Explanation

To solve the problem, we need to analyze the given vector equation:

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

### Step 1: Understand the vector triple product
The equation involves a triple product and a dot product. We can use the vector triple product identity, which states that:

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

### Step 2: Apply the identity
Let \(\vec{x} = \vec{a} \times \vec{b}\), \(\vec{y} = \vec{c}\), and \(\vec{z} = \vec{d}\). We can rewrite the expression:

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

### Step 3: Substitute back into the dot product
Now, substituting this back into the original equation gives us:

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

### Step 4: Expand the dot product
We can expand this dot product:

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

### Step 5: Analyze the conditions
For this equation to hold true, either:

1. \((\vec{a} \times \vec{b}) \cdot \vec{d} = 0\) or
2. \((\vec{c} \cdot (\vec{a} \times \vec{d})) = 0\)

This implies that either \(\vec{d}\) is in the plane formed by \(\vec{a}\) and \(\vec{b}\), or \(\vec{c}\) is in the plane formed by \(\vec{a}\) and \(\vec{d}\).

### Conclusion
Thus, we can conclude that:

- Option A: \(\vec{a}, \vec{b}, \vec{c}, \vec{d}\) are necessarily coplanar.
- Option B: \(\vec{b}\) lies in the plane of \(\vec{a}\) and \(\vec{d}\).
- Option C: \(\vec{c}\) lies in the plane of \(\vec{a}\) and \(\vec{d}\).

Therefore, options B, C, and D may be true.