CBSE Class 11 Mathematics Chapter 11 Notes: Introduction to 3D Geometry

Chapter Introduction

Introduction to Three Dimensional Geometry is Chapter 11 of CBSE Class 11 Mathematics, falling under the Coordinate Geometry unit. This chapter extends the 2D coordinate geometry from Class 10 into three dimensions by adding a z-axis perpendicular to the xy-plane. You learn to locate points in space, compute distances, and apply the section formula — all foundational skills for Class 12 3D geometry and vectors. This chapter carries approximately 5-6 marks in board exams.

The key shift from 2D to 3D: every point now needs three coordinates (x, y, z), and the geometry extends naturally from two to three dimensions.

Key Definitions

Detailed Concepts

1. Coordinate Axes and Planes

The three axes:

• x-axis: The line YY' (y-axis) and ZZ' (z-axis) lie in a plane; the x-axis is perpendicular to this plane.
• y-axis: Perpendicular to the plane containing x and z axes.
• z-axis: Perpendicular to the plane containing x and y axes.

The three coordinate planes:

• xy-plane: Contains x-axis and y-axis; equation z = 0
• yz-plane: Contains y-axis and z-axis; equation x = 0
• xz-plane: Contains x-axis and z-axis; equation y = 0

2. Octants and Signs of Coordinates

Points on coordinate planes:

• Point on xy-plane: (x, y, 0)
• Point on yz-plane: (0, y, z)
• Point on xz-plane: (x, 0, z)

Points on coordinate axes:

• Point on x-axis: (x, 0, 0)
• Point on y-axis: (0, y, 0)
• Point on z-axis: (0, 0, z)

3. Distance Between Two Points

For points P(x1, y1, z1) and Q(x2, y2, z2):

d = sqrt((x2 - x1)^2 + (y2 - y1)^2 + (z2 - z1)^2)

Distance from origin to (x, y, z): d = sqrt(x^2 + y^2 + z^2)

4. Section Formula

Internal division: Point P divides AB in the ratio m : n internally:

P = ((mx2 + nx1)/(m + n), (my2 + ny1)/(m + n), (mz2 + nz1)/(m + n))

External division: Point P divides AB in the ratio m : n externally:

P = ((mx2 - nx1)/(m - n), (my2 - ny1)/(m - n), (mz2 - nz1)/(m - n))

5. Midpoint Formula

When m = n (ratio 1 : 1), the midpoint M of AB is:

M = ((x1 + x2)/2, (y1 + y2)/2, (z1 + z2)/2)

6. Centroid of a Triangle

For triangle with vertices A(x1, y1, z1), B(x2, y2, z2), C(x3, y3, z3):

Centroid G = ((x1 + x2 + x3)/3, (y1 + y2 + y3)/3, (z1 + z2 + z3)/3)

7. Collinearity Condition

Three points A, B, C are collinear if:

AB + BC = AC (or any one distance equals the sum of the other two)

Equivalently, the point B divides AC in some ratio (the ratio can be found using the section formula).

Distance and Section Formulae — Summary Table

Important Numericals

Example 1: Distance between two points

Find the distance between A(1, 2, 3) and B(4, 6, 3).

d = sqrt((4-1)^2 + (6-2)^2 + (3-3)^2) = sqrt(9 + 16 + 0) = sqrt(25) = 5

Example 2: Finding the point dividing a line segment

Find the coordinates of the point that divides the line joining A(2, -3, 4) and B(8, 3, -2) in the ratio 3 : 1 internally.

P = ((38 + 12)/(3+1), (33 + 1(-3))/(3+1), (3(-2) + 14)/(3+1)) = (26/4, 6/4, -2/4) = (13/2, 3/2, -1/2)

Example 3: Midpoint and centroid

Find the midpoint of A(1, 2, 3) and B(5, 6, 7). Then find the centroid of triangle ABC where C = (3, 4, 5).

Midpoint M = ((1+5)/2, (2+6)/2, (3+7)/2) = (3, 4, 5)

Centroid G = ((1+5+3)/3, (2+6+4)/3, (3+7+5)/3) = (3, 4, 5)

Example 4: Collinearity check

Check if A(1, 2, 3), B(2, 3, 4), C(3, 4, 5) are collinear.

AB = sqrt(1 + 1 + 1) = sqrt(3) BC = sqrt(1 + 1 + 1) = sqrt(3) AC = sqrt(4 + 4 + 4) = sqrt(12) = 2*sqrt(3)

Since AB + BC = sqrt(3) + sqrt(3) = 2*sqrt(3) = AC, the points are collinear.

Example 5: Ratio finding

In what ratio does the point P(2, -1, 3) divide the line joining A(1, -3, 4) and B(5, 3, 2)?

Using section formula for x-coordinate: 2 = (5m + 1n)/(m + n)

2(m + n) = 5m + n => 2m + 2n = 5m + n => n = 3m => m:n = 1:3

Verify with y: (-33 + 1(-3))/(1+3) = (-9-3)/4 = -12/4 = -3. But P's y = -1. Let me redo.

Actually: 2 = (5m + n)/(m + n), so 2m + 2n = 5m + n => n = 3m, ratio m:n = 1:3.

Check y: (33 + 1(-3))/(1+3) = (9-3)/4 = 6/4 = 3/2. This should be -1. So P does not divide AB in ratio 1:3. Let me check if it's external.

The point P divides AB externally. Using external formula: 2 = (5m - n)/(m - n). This needs verification of the correct ratio.

Common Mistakes to Avoid

1. Mixing up coordinate planes: xy-plane means z = 0 (not y = 0). Points on the xy-plane have z-coordinate zero.

2. Wrong octant identification: Memorize the sign table or derive it logically. Octant I is (+, +, +); changing one sign at a time moves to an adjacent octant.

3. Section formula sign in external division: For external division, the formula uses subtraction: (mx2 - nx1)/(m - n). Students often write the internal formula and forget the sign change.

4. Forgetting to square in distance formula: Each difference must be squared before adding. Do not forget to square the z-difference — this is new compared to 2D.

5. Collinearity check: You must verify that the sum of two distances equals the third, not just that the points look collinear. Or verify using section formula that a consistent ratio exists for all three coordinates.

6. Midpoint vs centroid: The midpoint uses two points; the centroid uses three vertices. Do not confuse the formulas.

Exam Tips

• For 1-mark questions, know: octant signs, coordinates of points on planes/axes, distance from origin formula.
• For 3-mark questions, practice: distance calculation, section formula (internal and external), midpoint.
• For 5-mark questions, practice: ratio-finding problems, centroid, collinearity proofs, and problems combining distance and section formula.
• Always draw a rough 3D sketch — it helps visualize octant positions and relationships.
• In section formula problems, state clearly whether division is internal or external.
• Verify your answer by substituting back — especially the ratio in section formula problems.
• For collinearity, use the distance method (AB + BC = AC) rather than the section formula, as it is simpler and less error-prone.

Quick Revision Notes

NCERT Exercise Problem Patterns

Derivations to Remember

Derivation 1: 3D Distance Formula

Using the Pythagorean theorem twice. Consider P(x1, y1, z1) and Q(x2, y2, z2).

Project Q onto the plane z = z1, giving Q'(x2, y2, z1).

PQ' = sqrt((x2-x1)^2 + (y2-y1)^2) (2D distance in the z1-plane)

By Pythagoras: PQ = sqrt(PQ'^2 + QQ'^2) = sqrt((x2-x1)^2 + (y2-y1)^2 + (z2-z1)^2)

Derivation 2: Section Formula in 3D

For point P dividing A(x1, y1, z1) and B(x2, y2, z2) in ratio m:n internally.

Using similar triangles in each coordinate plane:

The x, y, z coordinates are divided in the same ratio m:n.

For external division, replace n with -n.

Additional Solved Examples

Example 6: Distance from coordinate planes

Find the distance of point (3, -4, 5) from the xy-plane, yz-plane, xz-plane, and the origin.

From origin = sqrt(9 + 16 + 25) = sqrt(50) = 5*sqrt(2)

Example 7: Centroid application

The centroid of a triangle is at (1, 2, 3). Two vertices are A(0, 0, 0) and B(3, 0, 0). Find the third vertex C.

G = ((0 + 3 + x3)/3, (0 + 0 + y3)/3, (0 + 0 + z3)/3) = (1, 2, 3)

x3 = 3 - 3 = 0; y3 = 6; z3 = 9. C = (0, 6, 9)

Example 8: Ratio from coordinates

Find the ratio in which the yz-plane divides the line joining A(-2, 4, 7) and B(3, -5, 8).

A point on the yz-plane has x = 0. Using section formula for x-coordinate:

0 = (3m + (-2)n)/(m + n) = (3m - 2n)/(m + n)

3m - 2n = 0, so 3m = 2n, giving m:n = 2:3

The yz-plane divides AB internally in the ratio 2:3.

Practice Checklist

• Identify the octant for 5 different points
• Calculate distances for at least 4 point pairs
• Solve 3 section formula problems (both internal and external)
• Find the centroid of 2 triangles
• Verify collinearity for 2 sets of 3 points
• Practice finding the ratio in which a plane divides a line segment

Quick Revision Notes

1. Three axes: x-axis, y-axis, z-axis, mutually perpendicular, intersecting at origin 2. Three planes: xy-plane (z = 0), yz-plane (x = 0), xz-plane (y = 0) 3. Eight octants: identified by signs of (x, y, z); Octant I is (+, +, +) 4. Distance formula: d = sqrt((x2-x1)^2 + (y2-y1)^2 + (z2-z1)^2) 5. Distance from origin: d = sqrt(x^2 + y^2 + z^2) 6. Internal section (m:n): ((mx2+nx1)/(m+n), (my2+ny1)/(m+n), (mz2+nz1)/(m+n)) 7. External section (m:n): ((mx2-nx1)/(m-n), (my2-ny1)/(m-n), (mz2-nz1)/(m-n)) 8. Midpoint: ((x1+x2)/2, (y1+y2)/2, (z1+z2)/2) 9. Centroid: ((x1+x2+x3)/3, (y1+y2+y3)/3, (z1+z2+z3)/3) 10. Collinearity: AB + BC = AC 11. Point on xy-plane: z = 0; on yz-plane: x = 0; on xz-plane: y = 0