Question

The angle between two lines in 3D space can be found using:

• A. Dot product of direction vectors
• B. Cross product only
• C. Determinant method
• D. Distance formula

Hint:

If the dot product of the direction vectors is zero, the two lines are perpendicular (\(90^\circ\)). This is a very common shortcut in 3D geometry problems!

Answer 1

The Correct Option is A

Step 1: Understanding the Concept
In 3D geometry, every line has a direction vector \(\vec{b} = a\hat{i} + b\hat{j} + c\hat{k}\), where \((a, b, c)\) are its direction ratios. The angle between two lines is essentially the angle between their direction vectors.

Step 2: Key Formula or Approach
If the direction vectors of two lines are \(\vec{b_1}\) and \(\vec{b_2}\), the angle \(\theta\) is given by the dot product formula: \[ \cos \theta = \frac{|\vec{b_1} \cdot \vec{b_2}|}{|\vec{b_1}| |\vec{b_2}|} \]

Step 3: Detailed Explanation
1. The dot product involves multiplying corresponding components and adding them: \( a_1a_2 + b_1b_2 + c_1c_2 \). 2. The cross product could give the sine of the angle, but the dot product is the standard and most direct way to find the cosine (and thus the angle) between any two vectors in space. 3. Distance formulas and determinants are used for other properties like the shortest distance between skew lines or finding volume, but not for finding angles directly.

Step 4: Final Answer
The angle is found using the dot product of direction vectors.