Three Dimensional Geometry Speed Test for Exam Prep - Interactive Mini Game Updated on Wednesday, August 7, 2025

Sharpen your Three Dimensional Geometry skills with this interactive mini-game designed to improve your speed and accuracy for competitive exams.

Updated: just now

Categories: Mini Game, Math, Class 11
Tags: Mini Game, Math, Class 11, 3D Geometry, Three Dimensional Geometry, Geometry
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Three Dimensional Geometry Cheatsheet & Quiz

Three Dimensional Geometry Cheatsheet

Key Concepts & Formulas

  • Point: Represented as \( P(x, y, z) \) in 3D space.
  • Distance Formula: \( d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2} \)
  • Midpoint Formula: \( \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}, \frac{z_1 + z_2}{2}\right) \)
  • Vector Form: \( \vec{AB} = (x_2 - x_1, y_2 - y_1, z_2 - z_1) \)
  • Dot Product: \( \vec{a} \cdot \vec{b} = a_1b_1 + a_2b_2 + a_3b_3 \)
  • Cross Product: \( \vec{a} \times \vec{b} = (a_2b_3 - a_3b_2, a_3b_1 - a_1b_3, a_1b_2 - a_2b_1) \)
  • Plane Equation: \( ax + by + cz + d = 0 \), where \( (a, b, c) \) is the normal vector.
  • Equation of a Line: Parametric form \( x = x_0 + at, y = y_0 + bt, z = z_0 + ct \)

Quick Reference Table

Concept Formula Description
Distance Between Points \( d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2} \) Euclidean distance between two points in 3D.
Midpoint \( \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}, \frac{z_1 + z_2}{2}\right) \) Midpoint of the line segment between two points.
Dot Product \( \vec{a} \cdot \vec{b} = a_1b_1 + a_2b_2 + a_3b_3 \) Measures angle and projection between vectors.
Cross Product \( \vec{a} \times \vec{b} = (a_2b_3 - a_3b_2, a_3b_1 - a_1b_3, a_1b_2 - a_2b_1) \) Vector perpendicular to two given vectors.
Plane Equation \( ax + by + cz + d = 0 \) General form of a plane in 3D.
Line Equation \( x = x_0 + at, y = y_0 + bt, z = z_0 + ct \) Parametric form of a line through a point.

Study Tips

Visualize: Draw diagrams to understand spatial relationships.

Master Vectors: Many 3D problems reduce to vector calculations.

Memorize Key Formulas: Distance, midpoint, dot & cross product, plane and line equations.

Practice Problems: Apply formulas to a variety of 3D coordinate geometry problems.

Check Work: Always verify by substituting points into plane or line equations.

Three Dimensional Geometry Quick Tips

  • Vector Basics: Understand vector addition, subtraction, and scalar multiplication.
  • Dot Product: Use for angle between vectors \( \cos \theta = \frac{\vec{a} \cdot \vec{b}}{|\vec{a}||\vec{b}|} \).
  • Cross Product: Use to find a vector perpendicular to two vectors.
  • Plane Identification: Use the normal vector to describe planes.
  • Line Equations: Use parametric or symmetric forms for lines in space.

Three Dimensional Geometry Speed Quiz

Test your speed with 5 three dimensional geometry questions! You have 30 seconds per question.