If `veca,vecb,vecc and vecd` are unit vectors such that `(vecaxxvecb).(veccxxvecd)=1 and veca.vecc=1/2` then (A) `veca,vecb,vecc` are non coplanar (B) `vecb,vecc, vecd` are non coplanar (C) `vecb, vecd` are non paralel (D) `veca, vecd` are paralel and `vecb, vecc` are parallel

Options

A. `veca, vecb and vecc` are non- coplanar
B. `vecb, vecc and vecd` are non-coplanar
C. `vecb and vecd` are non- parallel
D. `veca and vecd` are parallel and `vecb and vecc` are parallel

Explanation

To solve the problem, we need to analyze the given conditions involving the unit vectors \( \vec{a}, \vec{b}, \vec{c}, \) and \( \vec{d} \).

### Step 1: Analyze the first condition
The first condition given is:
\[ (\vec{a} \times \vec{b}) \cdot (\vec{c} \times \vec{d}) = 1 \]
Since \( \vec{a}, \vec{b}, \vec{c}, \) and \( \vec{d} \) are unit vectors, we can express the magnitudes of their cross products:
\[ |\vec{a} \times \vec{b}| = |\vec{a}| |\vec{b}| \sin \theta_1 = 1 \cdot 1 \cdot \sin \theta_1 = \sin \theta_1 \]
\[ |\vec{c} \times \vec{d}| = |\vec{c}| |\vec{d}| \sin \theta_2 = 1 \cdot 1 \cdot \sin \theta_2 = \sin \theta_2 \]
Thus, we can rewrite the first condition as:
\[ \sin \theta_1 \sin \theta_2 \cos \phi = 1 \]
where \( \phi \) is the angle between \( \vec{a} \times \vec{b} \) and \( \vec{c} \times \vec{d} \).

### Step 2: Determine the angles
Since \( \sin \theta_1 \) and \( \sin \theta_2 \) can only take values between 0 and 1, the only way for their product to equal 1 is if both \( \sin \theta_1 = 1 \) and \( \sin \theta_2 = 1 \), which implies:
\[ \theta_1 = 90^\circ \quad \text{and} \quad \theta_2 = 90^\circ \]
This means \( \vec{a} \) is perpendicular to \( \vec{b} \) and \( \vec{c} \) is perpendicular to \( \vec{d} \).

### Step 3: Analyze the angle \( \phi \)
Since \( \cos \phi = 1 \), we have:
\[ \phi = 0^\circ \]
This indicates that \( \vec{a} \times \vec{b} \) and \( \vec{c} \times \vec{d} \) are parallel.

### Step 4: Analyze the second condition
The second condition given is:
\[ \vec{a} \cdot \vec{c} = \frac{1}{2} \]
Let \( \theta_3 \) be the angle between \( \vec{a} \) and \( \vec{c} \):
\[ \vec{a} \cdot \vec{c} = |\vec{a}| |\vec{c}| \cos \theta_3 = 1 \cdot 1 \cdot \cos \theta_3 = \cos \theta_3 = \frac{1}{2} \]
This implies:
\[ \theta_3 = 60^\circ \quad \text{(or \( \frac{\pi}{3} \) radians)} \]

### Step 5: Visualize the vectors
Now we can visualize the vectors:
- \( \vec{a} \) and \( \vec{b} \) are perpendicular.
- \( \vec{c} \) and \( \vec{d} \) are perpendicular.
- The angle between \( \vec{a} \) and \( \vec{c} \) is \( 60^\circ \).

### Step 6: Determine coplanarity and parallelism
Since \( \vec{a} \) is perpendicular to \( \vec{b} \) and \( \vec{c} \) is perpendicular to \( \vec{d} \), and both pairs of vectors are in the same plane, we can conclude that:
- \( \vec{a}, \vec{b}, \vec{c} \) are coplanar.
- \( \vec{b}, \vec{c}, \vec{d} \) are also coplanar.

### Conclusion
From the analysis:
- \( \vec{b} \) and \( \vec{d} \) are not parallel because they can form angles of \( 60^\circ \) or \( 120^\circ \).
- \( \vec{a} \) and \( \vec{d} \) are not parallel either.

Thus, the correct option is:
(C) \( \vec{b}, \vec{d} \) are non-parallel.