Coplanarity and Parallelism Conditions on Unit Vectors

To determine the correct option, let us analyze the given conditions step-by-step:

1. Magnitudes of Cross Products:
   We are given that $\vec{a}, \vec{b}, \vec{c},$ and $\vec{d}$ are unit vectors, so:
   $|\vec{a}| = |\vec{b}| = |\vec{c}| = |\vec{d}| = 1$
   The magnitude of the cross product of two unit vectors satisfies:
   $|\vec{a} \times \vec{b}| = |\vec{a}||\vec{b}|\sin\theta_1 = \sin\theta_1 \le 1$
   $|\vec{c} \times \vec{d}| = |\vec{c}||\vec{d}|\sin\theta_2 = \sin\theta_2 \le 1$
   where $\theta_1$ is the angle between $\vec{a}$ and $\vec{b}$, and $\theta_2$ is the angle between $\vec{c}$ and $\vec{d}$.

2. Analysis of the Dot Product Equality:
   We are given:
   $(\vec{a} \times \vec{b}) \cdot (\vec{c} \times \vec{d}) = 1$
   By the Cauchy-Schwarz inequality for the dot product:
   $(\vec{a} \times \vec{b}) \cdot (\vec{c} \times \vec{d}) \le |\vec{a} \times \vec{b}| |\vec{c} \times \vec{d}| \le 1 \cdot 1 = 1$
   For the dot product to achieve its maximum possible value of $1$, the following must hold simultaneously:

   • $|\vec{a} \times \vec{b}| = 1 \implies \vec{a} \perp \vec{b}$
   • $|\vec{c} \times \vec{d}| = 1 \implies \vec{c} \perp \vec{d}$
   • The vectors $\vec{a} \times \vec{b}$ and $\vec{c} \times \vec{d}$ are in the same direction, meaning:
     $\vec{a} \times \vec{b} = \vec{c} \times \vec{d} = \hat{n}$
     where $\hat{n}$ is a common unit normal vector.

3. Coplanarity of the Vectors:
   Since $\vec{a} \times \vec{b} = \hat{n}$ and $\vec{c} \times \vec{d} = \hat{n}$, all four vectors $\vec{a}, \vec{b}, \vec{c},$ and $\vec{d}$ are orthogonal to the same unit vector $\hat{n}$. Consequently, all four vectors lie in the same plane perpendicular to $\hat{n}$.

   • Thus, any set of three vectors among them, such as $\{\vec{a}, \vec{b}, \vec{c}\}$ and $\{\vec{b}, \vec{c}, \vec{d}\}$, are coplanar.
   • This rules out options (A) and (B).

4. Relationship Between $\vec{b}$ and $\vec{d}$:
   Using the Lagrange's identity for the scalar product of cross products:
   $(\vec{a} \times \vec{b}) \cdot (\vec{c} \times \vec{d}) = (\vec{a} \cdot \vec{c})(\vec{b} \cdot \vec{d}) - (\vec{a} \cdot \vec{d})(\vec{b} \cdot \vec{c})$
   Alternatively, since all vectors lie in the same plane:

   • $\vec{b}$ is obtained by rotating $\vec{a}$ by an angle of $+90^\circ$ in the plane.
   • $\vec{d}$ is obtained by rotating $\vec{c}$ by the same angle of $+90^\circ$ in the plane.

   The angle between $\vec{b}$ and $\vec{d}$ is therefore identical to the angle between $\vec{a}$ and $\vec{c}$. Hence:
   $\vec{b} \cdot \vec{d} = \vec{a} \cdot \vec{c} = \frac{1}{2}$

5. Parallelism Condition:
   For unit vectors $\vec{b}$ and $\vec{d}$ to be parallel, their dot product must be $\pm 1$:
   $\vec{b} \cdot \vec{d} = \pm 1$
   Since $\vec{b} \cdot \vec{d} = \frac{1}{2} \ne \pm 1$, the vectors $\vec{b}$ and $\vec{d}$ are non-parallel.

   • Similarly, since $\vec{a} \cdot \vec{c} = \frac{1}{2}$, $\vec{a}$ and $\vec{c}$ are non-parallel, which also implies $\vec{a}$ and $\vec{d}$ cannot be parallel as $\vec{c} \perp \vec{d}$. This rules out option (D).

Therefore, $\vec{b}$ and $\vec{d}$ are non-parallel.

Correct Answer: C