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

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.