Class 11 Maths • Chapter 11 • Comprehensive Interactive Notes
In 3D geometry, the position of a point is defined by three coordinates \( (x, y, z) \). The three axes (X, Y, Z) divide space into 8 parts called Octants.
Input coordinates to find the Octant and position.
| Octant | X | Y | Z |
|---|---|---|---|
| I | + | + | + |
| II | - | + | + |
| III | - | - | + |
| IV | + | - | + |
| V | + | + | - |
| VI | - | + | - |
| VII | - | - | - |
| VIII | + | - | - |
| Location | Coordinates Form |
|---|---|
| X-axis | (x, 0, 0) |
| Y-axis | (0, y, 0) |
| Z-axis | (0, 0, z) |
| XY-plane | (x, y, 0) |
| YZ-plane | (0, y, z) |
| XZ-plane | (x, 0, z) |
Exam Note:
These are frequently tested in MCQs and short answers.
The distance between \( P(x_1, y_1, z_1) \) and \( Q(x_2, y_2, z_2) \) is:
\( PQ = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2 + (z_2-z_1)^2} \)
Point P (x1, y1, z1)
Point Q (x2, y2, z2)
If a point is \( P(x, y, z) \), then distance from origin \( O(0,0,0) \) is:
\[ OP = \sqrt{x^2 + y^2 + z^2} \]
CBSE Tip:
Often asked as a 1–2 mark direct question.
Coordinates of point R dividing PQ in ratio \( m:n \):
\( \left( \frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n}, \frac{mz_2 + nz_1}{m+n} \right) \)
Ratio m : n
(Uses P & Q points from above)
The centroid of a triangle with vertices \( (x_1,y_1,z_1), (x_2,y_2,z_2), (x_3,y_3,z_3) \) is:
\( G = \left( \frac{x_1+x_2+x_3}{3}, \frac{y_1+y_2+y_3}{3}, \frac{z_1+z_2+z_3}{3} \right) \)
Input 3 vertices to find the centroid.
1. Point (-2, 4, -7) lies in which octant?
2. Distance of point (3, 4, 5) from origin is:
3. The xy-plane has equation:
4. A point on the x-axis has coordinates of the form:
5. Midpoint of (2,3,4) and (4,1,-2) is:
Self-Test: