PYQ Boards Imp formulas | 3D Geometry Class 12 Chapter 11 Maths | CBSE NCERT with Solutions PDF

 

Important Concepts Covered in This Revision Lecture

(Chapter 11 – Three-Dimensional Geometry | Class 12 NCERT CBSE | Extremely Useful for 2026 Boards)

• Cartesian equation of a line$\frac{x - x_1}{a} = \frac{y - y_1}{b} = \frac{z - z_1}{c}$
• Vector form of a line$\vec{r} = \vec{a} + \lambda \vec{b}$
• Parametric form / Any point on the line Using parameter λ (or t, μ): $x = x_1 + a\lambda,\ y = y_1 + b\lambda,\ z = z_1 + c\lambda$
• Direction ratios (DRs) and Direction cosines (DCs) of a line
• Condition for two lines to be parallel$\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}$
• Condition for two lines to be perpendicular$a_1a_2 + b_1b_2 + c_1c_2 = 0$
• Direction ratios of line joining two points$(x_1, y_1, z_1)$ and $(x_2, y_2, z_2)$$x_2 - x_1,\ y_2 - y_1,\ z_2 - z_1$
• Mid-point formula in 3D$\left( \frac{x_1+x_2}{2},\ \frac{y_1+y_2}{2},\ \frac{z_1+z_2}{2} \right)$
• Foot of the perpendicular from a point to a line (most frequent 4–6 mark question)
• Image (reflection) of a point in a given line Key properties used:
  1. Foot of perpendicular Q is the mid-point of A and A'
  2. Line AA' is perpendicular to the given line → Solve using parametric point + perpendicular condition
• Equation of the line joining a point and its image (A and A')
• Distance of a point from a line = 7 units type questions → Use distance formula between point and general point on line → solve quadratic in λ → two possible points P1, P2
• Line perpendicular to two given lines (common normal) Direction ratios = $\vec{b_1} \times \vec{b_2}$ (vector cross product)
• Shortest distance between two skew lines (non-parallel, non-intersecting) $\mid \frac{(\overrightarrow{a_2}-\overrightarrow{a_1})\cdot (\overrightarrow{b_1}\times \overrightarrow{b_2})}{\mathrm{\mid }\overrightarrow{b_1}\times \overrightarrow{b_2}\mathrm{\mid }}$
• Equation of a line parallel to a given line and passing through a given point Same DRs, new point
• Conversion between Cartesian and vector forms of a line
• Case-study/application-based questions (metro lines, solar panels, pedestrian pathway) that combine parallel, perpendicular, shortest distance, and equation of line concepts

CBSE Class 12 Maths | 3D Geometry Chapter 11 Revision + 2025 Board Questions Full Solution | 2026 Boards Complete last-minute revision of Three Dimensional Geometry Class 12 CBSE Chapter 11 with all 2025 board paper questions solved step-by-step. Perfect for CBSE 2026 students targeting 95%+ in Maths! q1 Find the image A' of the point A(1, 6, 3) in the line x/1=y-1/2=z-2/3  Also, find the equation of the line joining A and A'. q2. Find the image A' of the point A(2, 1, 2) in the line l : 7 = 41 + 2j + 2k + 2 (i - j -k). Also, find the equation of line joining AA'. Find the foot of perpendicular from point A on the line l. q3 Find a point P on the line x+5/1=y+3/4=z-6/-9 from point Q(2, 4,-1) is 7 units. Also, find the equation of line joining P and Q. q4. Initially, two metro lines, Line A and Line B, each consisting of multiple stations are designed. The track for Line A is represented by Class 12 3d chapter cbse pdf Class 12 3D Geometry NCERT PDF 3D Geometry Class 12 NCERT Solutions PDF Class 12 3d chapter cbse solutions 3D Geometry Class 12 Notes PDF download 3D Geometry Class 12 PYQ boards PDF Class 12 3d chapter cbse notes Class 12 Maths Chapter 11 PDF 00:00 imp formulas 06:00 q1 13:38 q2 25:45 q3 34:30 q4 3d geometry class 12 three dimensional geometry class 12 class 12 maths chapter 11 3d geometry cbse 2026 3d geometry 2025 board paper image of point in a line foot of perpendicular 3d geometry shortest distance between skew lines cbse class 12 maths 2026 class 12 maths revision 3d geometry important questions direction ratios class 12 equation of line in 3d cbse 2025 maths solutions 3d geometry case study metro case study 3d geometry class 12 maths chapter 11 ncert 3d geometry one shot 3d geometry revision 2026 class 12 maths board exam 2026 cbse class 12 maths important questions 3d geometry pyq previous year questions 3d geometry class 12 maths chapter wise revision cbse maths 2026 strategy 3d geometry ncert solutions vector form of line class 12 cartesian form of line perpendicular lines in 3d parallel lines in 3d