Step 1: Understanding the Concept:
In 3D geometry, the orientation of a line is entirely defined by its direction vector (or direction ratios). The angle between two lines is essentially the angle between their respective direction vectors.
Step 2: Key Formula or Approach:
If $\vec{b_1}$ and $\vec{b_2}$ are the direction vectors of two lines, the angle $\theta$ between them is calculated using the dot product formula:
\[ \cos \theta = \frac{\vec{b_1} \cdot \vec{b_2}}{|\vec{b_1}| |\vec{b_2}|} \]
Step 3: Detailed Explanation:
Let's analyze the options:
Dot product: Directly yields $\cos \theta$. It is the standard, simplest, and most computationally efficient method to find the angle. It inherently handles the sign to determine if the angle is acute or obtuse.
Cross product: Yields $\sin \theta$ via $|\vec{b_1} \times \vec{b_2}| = |\vec{b_1}||\vec{b_2}|\sin\theta$. While possible, it's computationally heavier and ambiguous for obtuse angles since $\sin(\pi - \theta) = \sin\theta$. The option says "Cross product {only}", which is incorrect.
Determinant method / Distance formula: These are used for coplanarity, cross products, or shortest distance between skew lines, not for finding angles directly.
Step 4: Final Answer:
The angle is found using the dot product of direction vectors.