01 · Explore
The Need for Three Dimensional Geometry
While two-dimensional geometry helps locate points on a flat surface, real-world objects often exist in space, requiring a third dimension.
In plane geometry, we use two mutually perpendicular lines (x and y axes) to locate a point. However, to describe the position of a flying aeroplane or a bulb hanging from a ceiling, two coordinates are insufficient. We need to know the distances from two perpendicular walls and the height from the floor.
This leads to the requirement of three mutually perpendicular planes. The intersection of these planes provides three numbers, known as the coordinates of the point in space. This system allows us to precisely map any object in the physical world.
Pause & Try
Think it through first. Writing and checking your answer is free with an account.
Question
Why are two coordinates insufficient to locate a hanging light bulb in a room?
Sign in to see the answer
Write your own answer and compare it with ours. It’s free.
Sign inNew here? Sign up freeNCERT reference: chapter PDF page 1.
02 · Explore
Coordinate Axes and Coordinate Planes
The three-dimensional rectangular coordinate system is formed by three mutually perpendicular planes intersecting at a single point.
The point of intersection of the three planes is called the origin (O). The lines of intersection of these planes are the x-axis (X'OX), y-axis (Y'OY), and z-axis (Z'OZ). These axes are mutually perpendicular.
The three planes are named the XY-plane (containing x and y axes), the YZ-plane (containing y and z axes), and the ZX-plane (containing z and x axes). Distances are measured as positive or negative relative to these planes. For example, distances measured upwards from the XY-plane along the z-axis are positive, while downwards are negative.
Formation of the 3D Coordinate System
- 1
Origin (O)
The central point where three mutually perpendicular planes intersect.
- 2
Coordinate Axes
The lines X'OX, Y'OY, and Z'OZ formed by the intersection of the planes.
- 3
Coordinate Planes
The XY, YZ, and ZX planes that divide the space into sections.
- 4
Octants
The eight distinct regions of space created by the three intersecting planes.
The sequence of defining a 3D space starting from a single point of intersection to the division of space into eight octants.
Pause & Try
Think it through first. Writing and checking your answer is free with an account.
Question
What are the names of the three coordinate planes?
Sign in to see the answer
Write your own answer and compare it with ours. It’s free.
Sign inNew here? Sign up freeNCERT reference: chapter PDF page 2.
03 · Explore
Understanding Octants
The three coordinate planes divide the entire space into eight regions called octants.
In 2D geometry, axes divide a plane into four quadrants. In 3D geometry, the three planes divide space into eight octants, denoted by Roman numerals I to VIII. The sign of the x, y, and z coordinates determines which octant a point resides in.
For instance, in the first octant (I), all coordinates (x, y, z) are positive. In the fifth octant (V), x and y are positive, but z is negative. Understanding these signs is crucial for visualizing the position of a point.
| Octant | x-sign | y-sign | z-sign |
|---|---|---|---|
| I | + | + | + |
| II | – | + | + |
| III | – | – | + |
| IV | + | – | + |
| V | + | + | – |
| VI | – | + | – |
| VII | – | – | – |
| VIII | + | – | – |
Pause & Try
Think it through first. Writing and checking your answer is free with an account.
Question
In which octant does the point (4, -2, -5) lie?
Sign in to see the answer
Write your own answer and compare it with ours. It’s free.
Sign inNew here? Sign up freeNCERT reference: chapter PDF pages 2, 3.
04 · Explore
Coordinates of a Point in Space
Every point in space is uniquely identified by an ordered triplet of real numbers (x, y, z).
To find the coordinates of a point P, we drop a perpendicular to the XY-plane (point M). From M, we draw a perpendicular to the x-axis (point L). The distance OL is the x-coordinate, LM is the y-coordinate, and MP is the z-coordinate.
Alternatively, x, y, and z represent the perpendicular distances of point P from the YZ, ZX, and XY planes, respectively. This one-to-one correspondence ensures every point has a unique triplet and every triplet represents a unique point.
Coordinates on Axes and Planes
P = (x, 0, 0); Q = (0, y, z)
Any point lying on the x-axis has its y and z coordinates as zero. Similarly, any point lying in the YZ-plane has its x-coordinate as zero.
Pause & Try
Think it through first. Writing and checking your answer is free with an account.
Question
A point is in the XZ-plane. What is its y-coordinate?
Sign in to see the answer
Write your own answer and compare it with ours. It’s free.
Sign inNew here? Sign up freeNCERT reference: chapter PDF pages 2, 3, 4.
05 · Explore
The Distance Formula in 3D
The distance between two points in space is calculated using an extension of the Pythagorean theorem.
Consider two points P(x1, y1, z1) and Q(x2, y2, z2). By constructing a rectangular parallelepiped where PQ is the diagonal, we can apply the Pythagorean theorem in two steps.
First, we find the diagonal of the base in the horizontal plane, then use that result with the vertical height to find the spatial distance PQ. The resulting formula is the square root of the sum of the squares of the differences of the coordinates.
Calculating Distance Between Two Points
PQ = √((x2 - x1)² + (y2 - y1)² + (z2 - z1)²)
To find the distance between P(1, -3, 4) and Q(-4, 1, 2): 1. x2 - x1 = -4 - 1 = -5 2. y2 - y1 = 1 - (-3) = 4 3. z2 - z1 = 2 - 4 = -2 4. PQ = √((-5)² + 4² + (-2)²) = √(25 + 16 + 4) = √45 = 3√5 units.
Pause & Try
Think it through first. Writing and checking your answer is free with an account.
Question
Find the distance between (2, 3, 5) and (4, 3, 1).
Sign in to see the answer
Write your own answer and compare it with ours. It’s free.
Sign inNew here? Sign up freeNCERT reference: chapter PDF pages 4, 5.
06 · Explore
Collinearity and Geometric Properties
The distance formula can be used to verify if points are collinear or to identify types of triangles and quadrilaterals.
Three points A, B, and C are collinear if they lie on the same straight line. This is verified if the sum of the distances between two pairs of points equals the distance between the third pair (e.g., AB + BC = AC).
Similarly, we can verify if three points form a right-angled triangle by checking if the squares of the sides satisfy the Pythagoras theorem (a² + b² = c²). For a parallelogram, we check if opposite sides are equal in length. However, to distinguish a parallelogram from a rectangle, we must also check the diagonals; if the diagonals are unequal, the figure is a parallelogram but not a rectangle.
- 1
Calculate the distances between all pairs of points (AB, BC, and CA).
- 2
To check collinearity: See if the sum of two distances equals the third.
- 3
To check for a right triangle: See if the square of the longest side equals the sum of the squares of the other two sides.
- 4
To check for an isosceles triangle: See if any two side lengths are equal.
- 5
To distinguish a parallelogram from a rectangle: Verify opposite sides are equal, then calculate diagonals AC and BD. If AC ≠ BD, it is not a rectangle.
Pause & Try
Think it through first. Writing and checking your answer is free with an account.
Question
If PQ = √14, QR = 2√14, and PR = 3√14, are P, Q, and R collinear?
Sign in to see the answer
Write your own answer and compare it with ours. It’s free.
Sign inNew here? Sign up freeNCERT reference: chapter PDF pages 5, 6, 7.
07 · Explore
Centroid of a Triangle
The centroid is the point where the medians of a triangle intersect, representing the geometric center.
In 3D space, if a triangle has vertices A(x1, y1, z1), B(x2, y2, z2), and C(x3, y3, z3), the coordinates of its centroid G(x, y, z) are found by taking the average of the coordinates of the vertices.
This formula is a direct extension of the 2D centroid formula, adding the z-component average. It is useful in physics for finding the center of mass of a triangular plate.
Finding a Missing Vertex Using Centroid
x = (x1+x2+x3)/3, y = (y1+y2+y3)/3, z = (z1+z2+z3)/3
If centroid G is (1, 1, 1), A is (3, -5, 7), and B is (-1, 7, -6), find C(x, y, z): 1. (3 - 1 + x)/3 = 1 => 2 + x = 3 => x = 1 2. (-5 + 7 + y)/3 = 1 => 2 + y = 3 => y = 1 3. (7 - 6 + z)/3 = 1 => 1 + z = 3 => z = 2 C is (1, 1, 2).
Pause & Try
Think it through first. Writing and checking your answer is free with an account.
Question
What is the centroid of a triangle with vertices (0,0,6), (0,4,0), and (6,0,0)?
Sign in to see the answer
Write your own answer and compare it with ours. It’s free.
Sign inNew here? Sign up freeNCERT reference: chapter PDF pages 7, 8.
08 · Explore
Historical Development of 3D Geometry
The evolution of coordinate geometry from 2D to 3D involved several prominent mathematicians over centuries.
While René Descartes is the father of analytical geometry, his 1637 work primarily focused on 2D planes. The systematic development of 3D analytical geometry was later advanced by mathematicians like Antoinne Parent in 1700 and Leonhard Euler in 1748.
The introduction of the three coordinate planes we use today is attributed to J. Bernoulli in 1715. By the 19th century, these concepts were extended beyond three dimensions, eventually playing a vital role in Einstein's Theory of Relativity and the concept of the Space-Time Continuum.
Pause & Try
Think it through first. Writing and checking your answer is free with an account.
Question
Who is credited with the systematic development of analytical solid geometry in 1700?
Sign in to see the answer
Write your own answer and compare it with ours. It’s free.
Sign inNew here? Sign up freeNCERT reference: chapter PDF page 9.