Question

Let the pairs \( \mathbf{a}, \mathbf{b} \) and \( \mathbf{c}, \mathbf{d} \) each determine a plane. Then the planes are parallel if

• A. \( (\mathbf{a} \times \mathbf{c}) \cdot (\mathbf{b} \times \mathbf{d}) = 0 \)
• B. \( (\mathbf{a} \times \mathbf{c}) \cdot (\mathbf{b} \times \mathbf{d}) = 0 \)
• C. \( (\mathbf{a} \times \mathbf{b}) \cdot (\mathbf{c} \times \mathbf{d}) = 0 \)
• D. \( (\mathbf{a} \times \mathbf{b}) \cdot (\mathbf{c} \times \mathbf{d}) = 0 \)

Hint:

The condition for parallel planes is that the normal vectors must be parallel, which means the scalar triple product is zero.

Answer 1

The Correct Option is C

Step 1: Condition for parallel planes.
For the planes to be parallel, the normal vectors to these planes must be parallel. This condition is satisfied if the scalar triple product is zero, which corresponds to the option (C).

Step 2: Conclusion.
Thus, the correct answer is option (C).

Final Answer: \[ \boxed{\text{(C) } (\mathbf{a} \times \mathbf{b}) \cdot (\mathbf{c} \times \mathbf{d}) = 0} \]