Three Dimensional Geometry

Step 1: Understanding the Concept:
Direction ratios of a line are any three numbers which are proportional to the direction cosines of the line. For the axes, these represent the unit vector components along that specific axis.
Step 2: Detailed Explanation:
The x-axis makes an angle of 0° with itself, 90° with the y-axis, and 90° with the z-axis. The direction cosines ( ) are: Any numbers proportional to are direction ratios. The simplest form is .
Step 3: Final Answer:
The correct option is (b).

Concept: Line perpendicular to two given lines → direction vector = cross product of their direction vectors.
Step 1: Direction vectors.
Step 2: Cross product. Simplified direction:
Step 3: Equation of line:

Concept: Two lines are perpendicular if:
Step 1: Direction vectors.
Step 2: Apply dot product = 0. Scaling gives:

Concept: Angle between lines = angle between direction vectors:
Step 1: Direction vectors.
Step 2: Dot product.
Step 3: Magnitudes.
Step 4:

Concept: Area of triangle in 3D:
Step 1: Find vectors.
Step 2: Cross product.
Step 3: Magnitude.
Step 4: But matching options: