1st PUC Mathematics · Chapter 12
Introduction to Three Dimensional Geometry — Extending What You Already Know
This chapter is shorter and gentler than it sounds — almost everything in it is a direct extension of 2D coordinate geometry, with one extra axis and one extra coordinate added to every formula. Students who see it that way, instead of as an entirely new topic, move through it far faster.
The Three Axes and the Octants
In three dimensions, a point is located using three mutually perpendicular axes: x, y, and z. A point is written as (x, y, z), where each coordinate is the signed distance from the origin along that axis. Just as the two axes in 2D divide the plane into 4 quadrants, the three axes in 3D divide space into 8 octants, based on the sign combination of the three coordinates.
Distance Formula — One Extra Term
The 2D distance formula √[(x₂−x₁)² + (y₂−y₁)²] extends directly: the distance between (x₁,y₁,z₁) and (x₂,y₂,z₂) is √[(x₂−x₁)² + (y₂−y₁)² + (z₂−z₁)²] — literally the same formula with one more squared-difference term added under the root.
Worked example: Find the distance between (1, 2, 3) and (4, 6, 15). Distance = √[(4−1)² + (6−2)² + (15−3)²] = √[9 + 16 + 144] = √169 = 13.
Section Formula — Also One Extra Term
To find the point dividing the segment joining (x₁,y₁,z₁) and (x₂,y₂,z₂) in the ratio m:n internally, each coordinate follows the same pattern as 2D: x = (mx₂+nx₁)/(m+n), and identically for y and z. The midpoint is simply the special case where m = n = 1, giving the average of each coordinate pair.
Worked example: Find the point dividing the join of (2, 1, −3) and (5, −8, 3) in the ratio 2:1. x = (2·5 + 1·2)/3 = 12/3 = 4. y = (2·−8 + 1·1)/3 = −15/3 = −5. z = (2·3 + 1·−3)/3 = 3/3 = 1. The point is (4, −5, 1).
Where Students Actually Lose Marks
The most common error is arithmetic slips from tracking three coordinates instead of two — a sign error in the z-term is easy to miss when it’s the third repetition of a similar calculation. The second is confusing which point is (x₁,y₁,z₁) versus (x₂,y₂,z₂) in the section formula, which flips the ratio and gives the wrong point entirely.
Frequently Asked Questions
Is this chapter mostly new content or an extension of 2D geometry?
Almost entirely an extension — the underlying logic of distance and section formulas doesn’t change, only the number of coordinates being tracked.
Does 1st PUC cover planes and lines in 3D?
No — this introductory chapter covers only coordinates, distance and section formula. Full 3D Geometry with lines, planes and direction cosines is a 2nd PUC topic.
How does this connect to 2nd PUC?
Directly — 2nd PUC’s full Three Dimensional Geometry chapter builds on exactly these coordinate foundations, adding direction cosines, lines, and planes on top.
A Short Chapter Worth Getting Solid
Small now, foundational later — Daniel Sir makes sure this chapter’s basics are genuinely solid before 2nd PUC builds on them. See the PUC Mathematics Programme →