Class 11 Maths - ISC
Introduction to Three Dimensional Geometry
The Introduction to Three Dimensional Geometry chapter extends your understanding of coordinate geometry from two dimensions to three. In Class 11 ISC Mathematics, you will learn to locate points in space using ordered triplets (x, y, z), calculate distances between two points, find coordinates of section points, and determine the coordinates of centroids of triangles and tetrahedrons. This chapter builds a crucial foundation for vectors and calculus in 3D space, heavily featuring in ISC board exams as direct scoring questions, particularly distance formula applications and section formula derivations.
Start Learning FreeKey Concepts
Coordinate Axes and Planes
Space is divided into eight octants by three mutually perpendicular axes (X, Y, and Z) and three coordinate planes (XY, YZ, and ZX).
Coordinates of a Point in Space
Every point P in a three-dimensional Cartesian system is represented by an ordered triplet (x, y, z) denoting its perpendicular distances from the three coordinate planes.
Distance Formula
The straight-line distance between any two points P(x1, y1, z1) and Q(x2, y2, z2) in 3D space is given by the square root of the sum of squared differences of their coordinates.
Section Formula
The coordinates of a point dividing the join of two points in a given ratio m:n can be found for both internal and external division.
Centroid of a Triangle and Tetrahedron
The centroid coordinates are found by taking the arithmetic mean of the respective x, y, and z coordinates of all the vertices.
Important Formulas
Board Exam Info
In the ISC Class 11 Mathematics examination, this chapter typically carries around 4 to 6 marks. Common question types include finding the ratio in which a given line segment is divided by a coordinate plane, proving geometric properties like collinearity or vertices of specific triangles using the distance formula, and finding unknown coordinates.
Frequently Asked Questions
How do I determine which octant a point lies in?
You look at the signs of the x, y, and z coordinates. For example, a point with coordinates (+, -, +) lies in the fourth octant (OXYZ').
What is the difference between internal and external division formulas?
Internal division uses a plus sign (+) in the numerator and applies when the dividing point lies on the line segment. External division uses a minus sign (-) and applies when the point lies on the line extended outside the segment.
How do I prove that three points are collinear in 3D space?
You can use the distance formula to show that the sum of the lengths of two smaller segments equals the length of the largest segment (AB + BC = AC), or use ratio conditions.
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