Questions
2 questions in board exams
Difficulty
Easy
Importance
High yield for foundational marks
Overview
Three-Dimensional Geometry extends the principles of 2D Cartesian systems into space by introducing the z-axis. It is a fundamental topic for understanding spatial relationships and vector calculus, serving as a scoring bedrock for CBSE board exams and various engineering entrance papers.
Coordinates in Space
In 3D space, a point is represented by an ordered triplet (x, y, z), where x, y, and z are the perpendicular distances from the yz, xz, and xy planes respectively. Understanding the octants is crucial as they determine the signs of the coordinates.
- Coordinate axes: X-axis, Y-axis, Z-axis
- Coordinate planes: xy-plane (z=0), yz-plane (x=0), zx-plane (y=0)
- Point P in space: (x, y, z)
- There are 8 octants in 3D space
Distance Formula
The distance between two points in 3D space is an extension of the Pythagorean theorem. This formula is frequently used to prove geometric properties of shapes like triangles and quadrilaterals in 3D.
- Distance between P(x1, y1, z1) and Q(x2, y2, z2)
- Formula: d = sqrt((x2-x1)^2 + (y2-y1)^2 + (z2-z1)^2)
- Distance from origin (0,0,0) to P(x, y, z) is sqrt(x^2 + y^2 + z^2)
- Use this to check for collinearity by showing sum of two segments equals the third
Section Formula
The section formula calculates the coordinates of a point that divides a line segment in a given ratio. It is essential for problems involving midpoints, centroids of triangles, and internal/external division of segments.
- Internal division ratio m:n formula: ((mx2+nx1)/(m+n), (my2+ny1)/(m+n), (mz2+nz1)/(m+n))
- Midpoint formula: ((x1+x2)/2, (y1+y2)/2, (z1+z2)/2)
- Centroid of a triangle with vertices (x1, y1, z1), (x2, y2, z2), (x3, y3, z3): ((x1+x2+x3)/3, (y1+y2+y3)/3, (z1+z2+z3)/3)
- External division formula uses negative signs (m:n becomes m:-n)
Formula Sheet
d = sqrt((x2-x1)^2 + (y2-y1)^2 + (z2-z1)^2)
P = ((mx2+nx1)/(m+n), (my2+ny1)/(m+n), (mz2+nz1)/(m+n))
Midpoint = ((x1+x2)/2, (y1+y2)/2, (z1+z2)/2)
Centroid = ((x1+x2+x3)/3, (y1+y2+y3)/3, (z1+z2+z3)/3)
Exam Tip
Always draw a quick rough sketch or write down the coordinates clearly before applying the section formula to avoid sign errors, which are the leading cause of mark deduction.
Common Mistakes
- Forgetting to include the z-coordinate in calculations when transitioning from 2D logic.
- Mixing up internal and external division signs in the section formula.
- Neglecting the signs of coordinates when dealing with points in different octants.
More Revision Notes
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