Introduction to Three Dimensional Geometry | Class 11 Mathematics Notes

11.4 Distance between Two Points

This section extends the concept of distance between two points from two dimensions to three dimensions. Given two points P(x₁, y₁, z₁) and Q(x₂, y₂, z₂) in space, the distance PQ is derived using the Pythagorean theorem. By drawing planes through P and Q parallel to the coordinate planes, a rectangular parallelepiped is formed with PQ as the diagonal. Considering right triangles formed by these points, the square of the distance PQ is the sum of squares of the differences in x, y, and z coordinates. Hence, the distance formula is PQ = √[(x₂ - x₁)² + (y₂ - y₁)² + (z₂ - z₁)²]. In particular, if P is the origin O(0, 0, 0), then the distance OQ = √(x₂² + y₂² + z₂²). Several examples illustrate the use of this formula, including finding distances, checking collinearity of points by verifying if the sum of distances equals the third, and verifying if points form right-angled triangles by applying the Pythagorean theorem in three dimensions.

📊 Diagram: See figure_5: Fig 11.4 shows points P and Q with planes drawn parallel to coordinate planes forming a rectangular parallelepiped, illustrating the derivation of the distance formula.

🔗 Connection: This section's understanding of distance is essential for solving problems in exercises and for further study of geometric properties in three dimensions.