Quick Revision Notes

Class 11 Mathematics

Chapter: Introduction to Three Dimensional Geometry (त्रivee-विमीय ज्यामिति का परिचय)

1. Introduction to 3D Coordinate System

• Coordinate Axes: In 3D space, there are three mutually perpendicular axes called the X-axis, Y-axis, and Z-axis.
• Origin ($O$): The point of intersection of these three axes, denoted as $(0, 0, 0)$.
• Coordinate Planes: The three axes divide the space into three mutually perpendicular planes:
  • XY-plane: Equation is $z = 0$
  • YZ-plane: Equation is $x = 0$
  • ZX-plane: Equation is $y = 0$

2. Octants (अष्टांश)

The three coordinate planes divide the space into 8 parts known as Octants.

3. Distance Formula (दूरी सूत्र)

The distance between two points $P(x_1, y_1, z_1)$ and $Q(x_2, y_2, z_2)$ in 3D space is given by: $$PQ = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}$$

• Distance of a point $P(x, y, z)$ from the Origin $(0, 0, 0):$ $$OP = \sqrt{x^2 + y^2 + z^2}$$

4. Section Formula (विभाजन सूत्र)

Let $P(x_1, y_1, z_1)$ and $Q(x_2, y_2, z_2)$ be two points.

• Internal Division (आंतरिक विभाजन): The coordinates of the point $R(x, y, z)$ which divides the line segment $PQ$ in the ratio $m : n$ internally are: $$x = \frac{mx_2 + nx_1}{m + n}, \quad y = \frac{my_2 + ny_1}{m + n}, \quad z = \frac{mz_2 + nz_1}{m + n}$$

• External Division (बाह्य विभाजन): The coordinates of the point $R(x, y, z)$ which divides the line segment $PQ$ in the ratio $m : n$ externally are: $$x = \frac{mx_2 - nx_1}{m - n}, \quad y = \frac{my_2 - ny_1}{m - n}, \quad z = \frac{mz_2 - nz_1}{m - n}$$

• Mid-point Formula (मध्य-बिंदु सूत्र): The coordinates of the mid-point of $PQ$ (where ratio is $1:1$): $$\left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}, \frac{z_1 + z_2}{2}\right)$$

5. Centroid of a Triangle (त्रिभुज का केंद्रक)

If $A(x_1, y_1, z_1)$, $B(x_2, y_2, z_2)$, and $C(x_3, y_3, z_3)$ are the vertices of a triangle, then the coordinates of its centroid ($G$) are: $$G(x, y, z) = \left(\frac{x_1 + x_2 + x_3}{3}, \frac{y_1 + y_2 + y_3}{3}, \frac{z_1 + z_2 + z_3}{3}\right)$$

• Centroid of a Tetrahedron (चतुष्फलक का केंद्रक): If $(x_1, y_1, z_1), (x_2, y_2, z_2), (x_3, y_3, z_3), (x_4, y_4, z_4)$ are the vertices of a tetrahedron: $$\left(\frac{x_1 + x_2 + x_3 + x_4}{4}, \frac{y_1 + y_2 + y_3 + y_4}{4}, \frac{z_1 + z_2 + z_3 + z_4}{4}\right)$$

6. Important Points to Remember for MP Board Exams

1. Coordinates on Axes:
   • Any point on the X-axis is of the form $(x, 0, 0)$.
   • Any point on the Y-axis is of the form $(0, y, 0)$.
   • Any point on the Z-axis is of the form $(0, 0, z)$.
2. Coordinates on Planes:
   • Any point in the XY-plane is of the form $(x, y, 0)$.
   • Any point in the YZ-plane is of the form $(0, y, z)$.
   • Any point in the ZX-plane is of the form $(x, 0, z)$.
3. Collinearity: Three points $A, B, C$ are collinear if $AB + BC = AC$ (using distance formula) or if one is a scalar multiple of the section ratio.