If `((veca xx vec b ) xx (vec c xx vec d)).(vec a xx vec d)= 0` , then which of the following is alaways true ?

Options

A. `veca, vecb , vec c , vec d ` are necessarily coplanar
B. either `veca` or `vec d ` must lie in the plane of `vecb` and `vec c `
C. either `vecb` or `vec c ` must lie in the pane of `vec a ` and `vec d `
D. either `veca` or `vecb` must lie in the plane of `vec c ` and `vec d `

Answer

To solve the problem, we need to analyze the given expression and determine which condition is always true based on the provided equation.

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

### Step 1: Understand the expression
The expression involves cross products and a dot product. The equation states that the result of the left-hand side is equal to zero. This indicates that the vectors involved are somehow related in terms of their orientation in space.

### Step 2: Apply the vector triple product identity
We can use the vector triple product identity:
\[ \vec{x} \times (\vec{y} \times \vec{z}) = (\vec{x} \cdot \vec{z}) \vec{y} - (\vec{x} \cdot \vec{y}) \vec{z} \]
Let \(\vec{x} = \vec{a} \times \vec{b}\), \(\vec{y} = \vec{c}\), and \(\vec{z} = \vec{d}\). Thus, we can rewrite:
\[ (\vec{a} \times \vec{b}) \times (\vec{c} \times \vec{d}) = ((\vec{a} \times \vec{b}) \cdot \vec{d}) \vec{c} - ((\vec{a} \times \vec{b}) \cdot \vec{c}) \vec{d} \]

### Step 3: Substitute back into the equation
Substituting this back into the original equation gives:
\[ (((\vec{a} \times \vec{b}) \cdot \vec{d}) \vec{c} - ((\vec{a} \times \vec{b}) \cdot \vec{c}) \vec{d}) \cdot (\vec{a} \times \vec{d}) = 0 \]

### Step 4: Expand the dot product
Now, we can expand the 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 implications
For the above expression to equal zero, either:
1. \((\vec{a} \times \vec{b}) \cdot \vec{d} = 0\) or
2. \((\vec{c} \cdot (\vec{a} \times \vec{d}))\) must equal the ratio of the other term.

### Step 6: Conclusion
The condition \((\vec{a} \times \vec{b}) \cdot \vec{d} = 0\) implies that \(\vec{d}\) lies in the plane formed by \(\vec{a}\) and \(\vec{b}\). Similarly, the other terms imply relationships between \(\vec{c}\) and the plane formed by \(\vec{a}\) and \(\vec{d}\).

Thus, we conclude that:
- The vectors \(\vec{b}\) and \(\vec{c}\) must lie in the plane formed by \(\vec{a}\) and \(\vec{d}\).

### Final Answer
The correct option that is always true is:
- **Option 3**: \(\vec{b}\) and \(\vec{c}\) must lie in the plane of \(\vec{a}\) and \(\vec{d}\).