Core5 mapped NDA questions
Build the idea
Build exam-ready understanding by connecting the core idea, formulas or facts, worked example and previous NDA questions. Focus on why each rule works, then test it under timed conditions.
Concept
3D geometry extends coordinates to space (x,y,z). It deals with distances, direction cosines/ratios of a line, and equations of lines, planes and spheres. It is a frequent source of the paper's harder questions.
Direction cosines are the cosines of the angles a line makes with the axes.
Key Points & Formulas
- Distance from origin to (x,y,z) = √(x²+y²+z²).
- Direction cosines l,m,n satisfy l²+m²+n² = 1.
- Direction cosines = direction ratios divided by their magnitude.
- Plane: ax+by+cz+d = 0; (a,b,c) is the normal direction.
- Sphere centred (a,b,c) radius r: (x−a)²+(y−b)²+(z−c)² = r².
Worked Example
Q. Find the direction cosines of the line joining the origin to (2, 3, 6).
Magnitude = √(2²+3²+6²) = √(4+9+36) = √49 = 7.
Direction cosines = (2/7, 3/7, 6/7).
Check: (4+9+36)/49 = 1. ✓