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Planes

Study Guide

Key Definition

A plane is a two-dimensional flat surface that extends infinitely in all directions within three-dimensional space.

Important Notes

  • A plane in 3D space is defined by a normal vector and a point.
  • The equation of a plane: $A(x - x_0) + B(y - y_0) + C(z - z_0) = 0$
  • Alternatively, the general form: $Ax + By + Cz + D = 0$
  • The normal vector is perpendicular to the plane.
  • Understanding planes is crucial for solving 3D geometry problems.

Mathematical Notation

$+$ represents addition$-$ represents subtraction$\times$ represents multiplication$\div$ represents division$\angle ABC$ represents angle ABC$\cdot$ represents dot product$(x, y, z)$ represents a vector in 3D spaceRemember to use proper notation when solving problems

Why It Works

Planes are fundamental in 3D geometry as they allow us to work with surface equations and solve spatial problems using vectors.

Remember

A plane is defined by a normal vector and a point on the plane, using the formula $A(x - x_0) + B(y - y_0) + C(z - z_0) = 0$.

Quick Reference

Equation Form:$Ax + By + Cz + D = 0$

Understanding Planes

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Video explanation of this concept

concept. Use space or enter to play video.concept thumbnailBeginner

Start here! Easy to understand

BeginnerIntermediateAdvanced

Beginner Explanation

A plane is defined by a normal vector and a point in space, described by $n · (r - r_0) = 0$.Now showing Beginner level explanation.

Practice Problems

Test your understanding with practice problems

1

Quick Quiz

Single Choice QuizBeginner

Which of the following is a form of the equation of a plane? $A(x - x_0) + B(y - y_0) + C(z - z_0) = 0$

A$A(x - x_0) + B(y - y_0) + C(z - z_0) = 0$B$Ax + By = C$C$y = mx + b$D$x^2 + y^2 = r^2$Check AnswerPlease select an answer for all 1 questions before checking your answers. 1 question remaining.2

Real-World Problem

Question ExerciseIntermediate

Teenager Scenario

Imagine you want to build a skateboard ramp. The ramp surface is a plane described by the equation $2x + 3y - z = 6$. Calculate the acute angle θ between the ramp surface and the horizontal plane.Show AnswerClick to reveal the detailed solution for this question exercise.3

Thinking Challenge

Thinking ExerciseIntermediate

Think About This

Find the point where the plane $3x - 2y + z = 5$ intersects the x-axis.

Show AnswerClick to reveal the detailed explanation for this thinking exercise.4

Challenge Quiz

Single Choice QuizAdvanced

Which of the following vectors is perpendicular to the plane $4x - y + 2z = 8$?

A$(4, -1, 2)$B$(1, 0, 0)$C$(0, 1, 0)$D$(0, 0, 1)$Check AnswerPlease select an answer for all 1 questions before checking your answers. 1 question remaining.

Recap

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Review key concepts and takeaways

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