NCERT Class 11 Maths Chapter 11: Introduction to 3D Geometry

6 CBSE Marks

★★★★★ Difficulty

8 Topics

Medium JEE Weight

Topics Covered

8 key topics in this chapter

Coordinate Axes & Planes in 3D

Coordinates of a Point in Space

Octants of 3D Space

Distance Formula in 3D

Section Formula (Internal & External)

Midpoint in 3D

Distance from Coordinate Planes

Centroid of a Triangle in 3D

Study Resources

📖 NCERT Notes ✏️ MCQ Practice 📝 Exercise Solutions ✅ True / False Quiz

𝑓 Key Formulae

Essential mathematical expressions for this chapter — understand derivations, not just results.

Distance 3D

\[d = \sqrt{(x_2-x_1)^2+(y_2-y_1)^2+(z_2-z_1)^2}\]

📌 Extension of 2D distance formula

Midpoint

\[M = \left(\tfrac{x_1+x_2}{2},\tfrac{y_1+y_2}{2},\tfrac{z_1+z_2}{2}\right)\]

📌 Component-wise average

Section (int.)

\[P = \left(\tfrac{mx_2+nx_1}{m+n},\tfrac{my_2+ny_1}{m+n},\tfrac{mz_2+nz_1}{m+n}\right)\]

📌 P divides AB in ratio m:n internally

Section (ext.)

\[P = \left(\tfrac{mx_2-nx_1}{m-n},\tfrac{my_2-ny_1}{m-n},\tfrac{mz_2-nz_1}{m-n}\right)\]

📌 External division

Centroid

\[G = \left(\tfrac{x_1+x_2+x_3}{3},\tfrac{y_1+y_2+y_3}{3},\tfrac{z_1+z_2+z_3}{3}\right)\]

📌 Average of all three vertices

🎯 Exam-Ready Insights

Important points to remember — curated from CBSE Board question patterns.

CBSE mainly tests distance formula, section formula, and midpoint in 3D — master all three.

The three coordinate planes are xy (z=0), yz (x=0), and zx (y=0) — know which axis lies on which plane.

An octant is determined by the sign combination (+,+,+) to (−,−,−) of (x,y,z) — there are 8 octants.

Distance of point (x,y,z) from x-axis = √(y²+z²); from xy-plane = |z|.

Section formula in 3D works identically to 2D — just add the z-coordinate.

🏆 Competitive Exam Strategy

Targeted tips for JEE Main, JEE Advanced, NEET, BITSAT, and KVPY.

JEE Main

JEE Main treats 3D geometry heavily in Class XII (direction cosines, planes, lines) — Class XI is the foundation. Understand octants and axes thoroughly.

JEE Main

Collinearity in 3D: use the ratio condition — if P divides AB in ratio k:1, find k and check it is consistent across all three coordinates.

BITSAT

BITSAT tests distance from planes and axes — use the component-drop formula (e.g., distance from z-axis = √(x²+y²)).

KVPY

KVPY may test the centroid and circumcentre relationships in 3D — know that G divides the median in 2:1 ratio in 3D as well.

⚠️ Common Mistakes to Avoid

Forgetting the z-coordinate in 3D distance or section formulas.

Confusing the xy-plane (z=0) with the xz-plane (y=0).

Applying 2D centroid formula (2 vertices instead of 3) — centroid needs all three vertices.

Sign errors in external division formula (subtraction in denominator can catch students off-guard).

💡 Key Takeaways

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Three-dimensional coordinates (x, y, z) extend 2D geometry into space.

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The eight octants are determined by the signs of x, y, z.

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Distance and section formulas are direct coordinate-wise extensions of their 2D counterparts.

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This chapter is the gateway to vectors, 3D lines, and planes in Class XII.

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Every point on the x-axis has y = z = 0; similarly for y-axis and z-axis.

3-Dimensional Geometry