Kenneth ChengMathematical modeling
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Perpendicular lines and planes in space

Read the idea, work independently, then explain what changed.

高一必修 第二册(A版).pdf · 8.6 · PDF 153 / printed page 146

Revisit first: Parallel lines and planes in space

TOPIC 01

Perpendicular lines and planes in space

Build understanding of perpendicular lines and planes in space through definitions, contrasting cases and justified applications.

What you will be able to explain

  • Use definitions, valid spatial reasoning and metric checks for perpendicular lines and planes in space.
  • Connect representations and justify the steps, including boundary cases.
  • Explain a solution and apply the idea to a changed situation.

Line-plane test

Perpendicularity to two intersecting in-plane lines proves perpendicularity to the plane.

Angles and distances

Use orthogonal projections for line-plane angles and perpendicular sections for dihedral angles.

PREDICT → EXPLORE → EXPLAIN → TRANSFER

Make a prediction, then explore the relationship.

Lesson question: Before calculating, predict how the conclusion changes when one defining condition in perpendicular lines and planes in space changes. Record a reason.

Model exploration: predict which displayed result changes with the parameters, check the values, and compare the observation with the lesson question.

Schematic cuboid (not to scale): 3×4×2; volume=24, surface area=52.

Schematic cuboid (not to scale): 3×4×2; volume=24, surface area=52.

Explain: Compare two admissible cases and one boundary or invalid case. Explain the observed difference using the stated definition.

Transfer: Construct a new example and a tempting incorrect solution. Repair the solution by naming the missing condition.

Try it. Leave your reasoning visible.

Use one hint at a time. A correction explains what changed, not just the final answer.

01 / Foundation#Worked example

State a line-plane perpendicularity criterion.

  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Use perpendicularity tests with intersecting directions, then identify the relevant projection or section.
Hint 2
Two parallel test lines give only one direction.
Worked solution

  1. Use perpendicularity tests with intersecting directions, then identify the relevant projection or section.

  2. Calculate or simplify this relation.

    l⊥a, l⊥b,a,b⊆α,a∩b≠∅⇒l⊥αl\perp a,\ l\perp b,\quad a,b\subseteq\alpha,\quad a\cap b\ne\varnothing\Rightarrow l\perp\alpha

  3. The two intersecting directions span the plane.

A line perpendicular to two intersecting lines in a plane is perpendicular to that plane.

Checks and common pitfalls: The two intersecting directions span the plane.

Reasoning checklist · self / teacher assessment
  • State the relevant definition, condition or model.
  • Show a valid calculation, proof or counterexample.
  • Interpret the conclusion with its restrictions.

Think first. Reveal a hint when the class is ready.

02 / Standard#Worked example

Find the distance from P to the plane.

P=(2,2,5),α:z=0P=(2,2,5),\quad\alpha:z=0
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Use perpendicularity tests with intersecting directions, then identify the relevant projection or section.
Hint 2
Drop a perpendicular parallel to the z-axis.
Worked solution

  1. Use perpendicularity tests with intersecting directions, then identify the relevant projection or section.

  2. Calculate or simplify this relation.

    H=(2,2,0),PH=5H=(2,2,0),\quad PH=5

  3. Distance to a plane is the perpendicular length, not distance to the origin.

The requested value is 5.

Checks and common pitfalls: Distance to a plane is the perpendicular length, not distance to the origin.

Think first. Reveal a hint when the class is ready.

03 / Transfer#Worked example

Explain the projection criterion: a line in a plane is perpendicular to an oblique line exactly when it is perpendicular to its nonzero orthogonal projection.

v=p+n\mathbf v=\mathbf p+\mathbf n
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Use perpendicularity tests with intersecting directions, then identify the relevant projection or section.
Hint 2
Resolve the oblique vector into in-plane and normal components.
Worked solution

  1. Use perpendicularity tests with intersecting directions, then identify the relevant projection or section.

  2. Calculate or simplify this relation.

    u⋅v=u⋅p+u⋅n=u⋅p\mathbf u\cdot\mathbf v=\mathbf u\cdot\mathbf p+\mathbf u\cdot\mathbf n=\mathbf u\cdot\mathbf p

  3. A nonzero projection is required to speak of its direction.

The normal component has zero dot product with every in-plane direction, leaving the projection component.

Checks and common pitfalls: A nonzero projection is required to speak of its direction.

Reasoning checklist · self / teacher assessment
  • State the relevant definition, condition or model.
  • Show a valid calculation, proof or counterexample.
  • Interpret the conclusion with its restrictions.

Think first. Reveal a hint when the class is ready.

04 / Foundation#Your turn

Show why perpendicularity to one line in a plane is insufficient to prove line-plane perpendicularity.

v=(0,1,1),α:z=0\mathbf v=(0,1,1),\quad\alpha:z=0
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Test two independent in-plane directions.
Hint 2
A normal must be perpendicular to both.
Worked solution

  1. Test two independent in-plane directions.

  2. Calculate or simplify this relation.

    (0,1,1)⋅(1,0,0)=0;(0,1,1)⋅(0,1,0)=1(0,1,1)\cdot(1,0,0)=0;\quad(0,1,1)\cdot(0,1,0)=1

  3. The counterexample isolates the missing second direction.

v is perpendicular to the x-axis but not to the y-axis, so it is not normal to α.

Checks and common pitfalls: The counterexample isolates the missing second direction.

Reasoning checklist · self / teacher assessment
  • State the relevant definition, condition or model.
  • Show a valid calculation, proof or counterexample.
  • Interpret the conclusion with its restrictions.

Think first. Reveal a hint when the class is ready.

05 / Foundation#Your turn

Find the distance from P to the plane.

P=(3,2,6),α:z=0P=(3,2,6),\quad\alpha:z=0
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Use perpendicularity tests with intersecting directions, then identify the relevant projection or section.
Hint 2
Drop a perpendicular parallel to the z-axis.
Worked solution

  1. Use perpendicularity tests with intersecting directions, then identify the relevant projection or section.

  2. Calculate or simplify this relation.

    H=(3,2,0),PH=6H=(3,2,0),\quad PH=6

  3. Distance to a plane is the perpendicular length, not distance to the origin.

The requested value is 6.

Checks and common pitfalls: Distance to a plane is the perpendicular length, not distance to the origin.

Think first. Reveal a hint when the class is ready.

06 / Foundation#Your turn

A segment makes 30° with a plane and has length 2k. Find its perpendicular component.

k=3k=3
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Use perpendicularity tests with intersecting directions, then identify the relevant projection or section.
Hint 2
The line-plane angle is measured against the orthogonal projection.
Worked solution

  1. Use perpendicularity tests with intersecting directions, then identify the relevant projection or section.

  2. Calculate or simplify this relation.

    h=2(3)sin⁡30∘=3h=2(3)\sin30^{\circ}=3

  3. The perpendicular component uses sine of that angle.

The requested value is 3.

Checks and common pitfalls: The perpendicular component uses sine of that angle.

Think first. Reveal a hint when the class is ready.

07 / Foundation#Your turn

Find sin²θ for a cube’s space diagonal making angle θ with the base.

  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Use perpendicularity tests with intersecting directions, then identify the relevant projection or section.
Hint 2
Use vertical height over diagonal length.
Worked solution

  1. Use perpendicularity tests with intersecting directions, then identify the relevant projection or section.

  2. Calculate or simplify this relation.

    sin⁡θ=aa3=13;sin⁡2θ=1/3\sin\theta=\frac{a}{a\sqrt3}=\frac1{\sqrt3};\quad\sin^2\theta=1/3

  3. The cube size cancels, so the angle is scale-independent.

The requested value is 0.33333333.

Checks and common pitfalls: The cube size cancels, so the angle is scale-independent.

Think first. Reveal a hint when the class is ready.

08 / Standard#Your turn

Do perpendicular planes force every line in one plane to be perpendicular to the other? Give a counterexample.

  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Use perpendicularity tests with intersecting directions, then identify the relevant projection or section.
Hint 2
Consider the common intersection line.
Worked solution

  1. Use perpendicularity tests with intersecting directions, then identify the relevant projection or section.

  2. Calculate or simplify this relation.

    α:x=0,β:y=0,l:x=y=0\alpha:x=0,\quad\beta:y=0,\quad l:x=y=0

  3. A line in one plane must also be perpendicular to the intersection line to be normal to the other plane.

No: their intersection line lies in both planes and is not perpendicular to either.

Checks and common pitfalls: A line in one plane must also be perpendicular to the intersection line to be normal to the other plane.

Reasoning checklist · self / teacher assessment
  • State the relevant definition, condition or model.
  • Show a valid calculation, proof or counterexample.
  • Interpret the conclusion with its restrictions.

Think first. Reveal a hint when the class is ready.

09 / Standard#Your turn

Find the dihedral angle between the coordinate planes x=0 and y=0.

  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Use perpendicularity tests with intersecting directions, then identify the relevant projection or section.
Hint 2
Use a cross-section perpendicular to their common z-axis.
Worked solution

  1. Use perpendicularity tests with intersecting directions, then identify the relevant projection or section.

  2. Calculate or simplify this relation.

    n1=(1,0,0),n2=(0,1,0),n1⋅n2=0\mathbf n_1=(1,0,0),\quad\mathbf n_2=(0,1,0),\quad\mathbf n_1\cdot\mathbf n_2=0

  3. The section gives two perpendicular coordinate axes.

The requested value is 90.

Checks and common pitfalls: The section gives two perpendicular coordinate axes.

Think first. Reveal a hint when the class is ready.

10 / Standard#Your turn

In two perpendicular planes α and β, a line m in α is perpendicular to their intersection line. State its relation to β and justify by a perpendicular cross-section.

  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Take the section through m perpendicular to the common line l.
Hint 2
The right dihedral angle becomes a right angle between section lines.
Worked solution

  1. Take the section through m perpendicular to the common line l.

  2. Calculate or simplify this relation.

    m⊥l,n⊥l,∠(m,n)=90∘m\perp l,\quad n\perp l,\quad\angle(m,n)=90^{\circ}

  3. m is perpendicular to two intersecting directions in β, namely l and n.

m is perpendicular to β.

Checks and common pitfalls: m is perpendicular to two intersecting directions in β, namely l and n.

Reasoning checklist · self / teacher assessment
  • State the relevant definition, condition or model.
  • Show a valid calculation, proof or counterexample.
  • Interpret the conclusion with its restrictions.

Think first. Reveal a hint when the class is ready.

11 / Standard#Your turn

If l is perpendicular to α and m is parallel to l, prove m is perpendicular to α.

  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Consider every in-plane direction u.
Hint 2
A scalar multiple of a normal is still a normal when nonzero.
Worked solution

  1. Consider every in-plane direction u.

  2. Calculate or simplify this relation.

    m=cl,l⋅u=0⇒m⋅u=0\mathbf m=c\mathbf l,\quad\mathbf l\cdot\mathbf u=0\Rightarrow\mathbf m\cdot\mathbf u=0

  3. The locations of the lines do not affect their direction-based perpendicularity.

Parallel lines have proportional nonzero direction vectors, so the same normal direction is retained.

Checks and common pitfalls: The locations of the lines do not affect their direction-based perpendicularity.

Reasoning checklist · self / teacher assessment
  • State the relevant definition, condition or model.
  • Show a valid calculation, proof or counterexample.
  • Interpret the conclusion with its restrictions.

Think first. Reveal a hint when the class is ready.

12 / Transfer#Your turn

Find the distance from P to the plane.

P=(4,2,7),α:z=0P=(4,2,7),\quad\alpha:z=0
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Use perpendicularity tests with intersecting directions, then identify the relevant projection or section.
Hint 2
Drop a perpendicular parallel to the z-axis.
Worked solution

  1. Use perpendicularity tests with intersecting directions, then identify the relevant projection or section.

  2. Calculate or simplify this relation.

    H=(4,2,0),PH=7H=(4,2,0),\quad PH=7

  3. Distance to a plane is the perpendicular length, not distance to the origin.

The requested value is 7.

Checks and common pitfalls: Distance to a plane is the perpendicular length, not distance to the origin.

Think first. Reveal a hint when the class is ready.

13 / Transfer#Your turn

A segment makes 30° with a plane and has length 2k. Find its perpendicular component.

k=4k=4
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Use perpendicularity tests with intersecting directions, then identify the relevant projection or section.
Hint 2
The line-plane angle is measured against the orthogonal projection.
Worked solution

  1. Use perpendicularity tests with intersecting directions, then identify the relevant projection or section.

  2. Calculate or simplify this relation.

    h=2(4)sin⁡30∘=4h=2(4)\sin30^{\circ}=4

  3. The perpendicular component uses sine of that angle.

The requested value is 4.

Checks and common pitfalls: The perpendicular component uses sine of that angle.

Think first. Reveal a hint when the class is ready.

Focus on one question

End-of-lesson check and correction

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    Teacher preparation and assessment

    Question sequence

    • What must be true before using the main rule for perpendicular lines and planes in space?
    • Which representation makes this task easier, and why?
    • Change one assumption. Does the conclusion survive?

    Board plan

    • Use definitions, valid spatial reasoning and metric checks for perpendicular lines and planes in space.
    • Line-plane test: Perpendicularity to two intersecting in-plane lines proves perpendicularity to the plane.
    • Angles and distances: Use orthogonal projections for line-plane angles and perpendicular sections for dihedral angles.
    • Close with: conditions → representation → reasoning → check.

    Anticipated thinking

    • Expected reasoning: Perpendicularity to two intersecting in-plane lines proves perpendicularity to the plane.
    • Expected reasoning: Use orthogonal projections for line-plane angles and perpendicular sections for dihedral angles.
    • Expected correction: Perpendicular planes do not make every contained line perpendicular to the other plane.

    Assessment checklist

    • 1: identify the givens and required quantity.
    • 1: choose a valid definition, representation or method.
    • 1: present connected, correct reasoning.
    • 1: check conditions and explain the result.

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    Curriculum and source notes ↗