Question

If \( \vec{a} \times \vec{b} \) and \( \vec{c} \times \vec{d} \) are perpendicular, then which of the following is always true?

• A. \( \vec{a}, \vec{b}, \vec{c}, \vec{d} \) are necessarily coplanar
• B. Either \( \vec{a} \cdot \vec{d} \) must lie in the plane of \( \vec{b} \) and \( \vec{c} \)
• C. Either \( \vec{a} \cdot \vec{b} \) must lie in the plane of \( \vec{c} \) and \( \vec{d} \)
• D. Either \( \vec{a} \cdot \vec{b} \) must lie in the plane of \( \vec{c} \) and \( \vec{d} \)

Hint:

Two perpendicular cross products always lie on the same plane, making the vectors coplanar.

Answer 1

The Correct Option is A

Step 1: Apply the condition for perpendicular vectors.
If two vectors are perpendicular, their cross products are coplanar, meaning \( \vec{a}, \vec{b}, \vec{c}, \vec{d} \) lie on the same plane.
Step 2: Conclusion.
Thus, the vectors are necessarily coplanar.
Final Answer: \[ \boxed{\text{The vectors are necessarily coplanar.}} \]