Kenneth ChengMathematical modeling
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LEARN · EXPLAIN · REVISE

Spatial lines, planes, spheres and cross products

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

T06高三理組數學思維本(2026).pdf · Spatial analytic geometry; also PDF10–11 cross product · PDF 15 / printed page 14

Revisit first: Applications of space vectors

TOPIC 01

Spatial lines, planes, spheres and cross products

Construct spatial equations and use normals, cross products and direction tests.

What you will be able to explain

  • Construct spatial equations and use normals, cross products and direction tests.
  • Justify the method and check the conditions in a new situation.

Model or definition

Construct spatial equations and use normals, cross products and direction tests.

ℓ:P+su;Π:n⋅(X−P)=0\ell:P+su;\quad\Pi:n\cdot(X-P)=0

Conditions

Directions and normals must be nonzero; parallel directions do not prove that two spatial lines coincide.

PREDICT → EXPLORE → EXPLAIN → TRANSFER

Make a prediction, then explore the relationship.

Lesson question: Can two nonparallel space lines have no intersection?

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

Space vector v=(2,1,1); |v|=2.4495; v·(2,1,3)=8. The drawing shows the xy-projection, not its full spatial length.

Space vector v=(2,1,1); |v|=2.4495; v·(2,1,3)=8. The drawing shows the xy-projection, not its full spatial length.

Explain: Compare two admissible cases and explain their different results using the stated model.

Transfer: Use all three coordinate equations to distinguish intersecting and skew lines.

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

Find the plane through (t,0,0) with normal (1,2,3).

t=2t=2
  • Directions and normals must be nonzero; parallel directions do not prove that two spatial lines coincide.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Write the defining equation, retain exclusions, and then solve or compare.
Hint 2
Use this intermediate relation.
n⋅(X−P)=0n·(X−P)=0
Worked solution

  1. Write the defining equation, retain exclusions, and then solve or compare.

  2. Apply the stated relation and retain its conditions.

    (x−2)+2y+3z=0(x-2)+2y+3z=0

  3. The point fixes the constant and the normal fixes the coefficients.

The requested relation or conclusion is shown below.

x+2y+3z=2x+2y+3z=2

Checks and common pitfalls: The point fixes the constant and the normal fixes the coefficients.

Reasoning checklist · self / teacher assessment
  • State a valid definition or model and its assumptions.
  • Show the intermediate mathematical relations, not only the final claim.
  • Check exclusions, units or the interpretation of the result.

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

02 / Standard#Worked example

Calculate u×v.

u=(3,0,0),v=(0,2,0)u=(3,0,0),\quad v=(0,2,0)
  • Directions and normals must be nonzero; parallel directions do not prove that two spatial lines coincide.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Write the defining equation, retain exclusions, and then solve or compare.
Hint 2
Use the oriented determinant formula.
Worked solution

  1. Write the defining equation, retain exclusions, and then solve or compare.

  2. Apply the stated relation and retain its conditions.

    u×v=(0,0,6)u\times v=(0,0,6)

  3. Reversing the order changes the cross product sign.

The requested relation or conclusion is shown below.

u×v=(0,0,6)u\times v=(0,0,6)

Checks and common pitfalls: Reversing the order changes the cross product sign.

Reasoning checklist · self / teacher assessment
  • State a valid definition or model and its assumptions.
  • Show the intermediate mathematical relations, not only the final claim.
  • Check exclusions, units or the interpretation of the result.

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

03 / Transfer#Worked example

Classify the two spatial lines.

ℓ1=(s,0,0),ℓ2=(0,u,4)\ell_1=(s,0,0),\quad\ell_2=(0,u,4)
  • Directions and normals must be nonzero; parallel directions do not prove that two spatial lines coincide.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Write the defining equation, retain exclusions, and then solve or compare.
Hint 2
Equate all three coordinates.
Worked solution

  1. Write the defining equation, retain exclusions, and then solve or compare.

  2. Apply the stated relation and retain its conditions.

    s=0, u=0, 0=4 impossibles=0,\ u=0,\ 0=4\text{ impossible}

  3. Their directions are not parallel but their z-levels prevent intersection.

The requested relation or conclusion is shown below.

skew lines\text{skew lines}

Checks and common pitfalls: Their directions are not parallel but their z-levels prevent intersection.

Reasoning checklist · self / teacher assessment
  • State a valid definition or model and its assumptions.
  • Show the intermediate mathematical relations, not only the final claim.
  • Check exclusions, units or the interpretation of the result.

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

04 / Foundation#Your turn

Find the plane through (t,0,0) with normal (1,2,3).

t=5t=5
  • Directions and normals must be nonzero; parallel directions do not prove that two spatial lines coincide.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Write the defining equation, retain exclusions, and then solve or compare.
Hint 2
Use this intermediate relation.
n⋅(X−P)=0n·(X−P)=0
Worked solution

  1. Write the defining equation, retain exclusions, and then solve or compare.

  2. Apply the stated relation and retain its conditions.

    (x−5)+2y+3z=0(x-5)+2y+3z=0

  3. The point fixes the constant and the normal fixes the coefficients.

The requested relation or conclusion is shown below.

x+2y+3z=5x+2y+3z=5

Checks and common pitfalls: The point fixes the constant and the normal fixes the coefficients.

Reasoning checklist · self / teacher assessment
  • State a valid definition or model and its assumptions.
  • Show the intermediate mathematical relations, not only the final claim.
  • Check exclusions, units or the interpretation of the result.

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

05 / Foundation#Your turn

Find the sphere radius.

(x−1)2+(y+2)2+(z−6)2=9(x-1)^2+(y+2)^2+(z-6)^2=9
  • Directions and normals must be nonzero; parallel directions do not prove that two spatial lines coincide.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Write the defining equation, retain exclusions, and then solve or compare.
Hint 2
Use this intermediate relation.
r2=9r²=9
Worked solution

  1. Write the defining equation, retain exclusions, and then solve or compare.

  2. Apply the stated relation and retain its conditions.

    r=3r=3

  3. A sphere uses three squared coordinate displacements.

The requested value is 3.

Checks and common pitfalls: A sphere uses three squared coordinate displacements.

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

06 / Foundation#Your turn

Find the line through P with direction u.

P=(1,2,7),u=(2,−1,3)P=(1,2,7),\quad u=(2,-1,3)
  • Directions and normals must be nonzero; parallel directions do not prove that two spatial lines coincide.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Write the defining equation, retain exclusions, and then solve or compare.
Hint 2
Use this intermediate relation.
X=P+suX=P+su
Worked solution

  1. Write the defining equation, retain exclusions, and then solve or compare.

  2. Apply the stated relation and retain its conditions.

    x=1+2s, y=2−s, z=7+3sx=1+2s,\ y=2-s,\ z=7+3s

  3. One parameter controls all coordinates simultaneously.

The requested relation or conclusion is shown below.

(x,y,z)=(1,2,7)+s(2,−1,3)(x,y,z)=(1,2,7)+s(2,-1,3)

Checks and common pitfalls: One parameter controls all coordinates simultaneously.

Reasoning checklist · self / teacher assessment
  • State a valid definition or model and its assumptions.
  • Show the intermediate mathematical relations, not only the final claim.
  • Check exclusions, units or the interpretation of the result.

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

07 / Foundation#Your turn

The line X=(0,0,t)+s(1,0,−1) meets z=0. Find s.

t=8t=8
  • Directions and normals must be nonzero; parallel directions do not prove that two spatial lines coincide.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Write the defining equation, retain exclusions, and then solve or compare.
Hint 2
Use this intermediate relation.
z=t−s=0z=t−s=0
Worked solution

  1. Write the defining equation, retain exclusions, and then solve or compare.

  2. Apply the stated relation and retain its conditions.

    s=8s=8

  3. Substitute the resulting parameter into every coordinate to obtain the intersection.

The requested value is 8.

Checks and common pitfalls: Substitute the resulting parameter into every coordinate to obtain the intersection.

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

08 / Standard#Your turn

The line X=(0,0,t)+s(1,0,−1) meets z=0. Find s.

t=9t=9
  • Directions and normals must be nonzero; parallel directions do not prove that two spatial lines coincide.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Write the defining equation, retain exclusions, and then solve or compare.
Hint 2
Use this intermediate relation.
z=t−s=0z=t−s=0
Worked solution

  1. Write the defining equation, retain exclusions, and then solve or compare.

  2. Apply the stated relation and retain its conditions.

    s=9s=9

  3. Substitute the resulting parameter into every coordinate to obtain the intersection.

The requested value is 9.

Checks and common pitfalls: Substitute the resulting parameter into every coordinate to obtain the intersection.

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

09 / Standard#Your turn

Calculate u×v.

u=(10,0,0),v=(0,2,0)u=(10,0,0),\quad v=(0,2,0)
  • Directions and normals must be nonzero; parallel directions do not prove that two spatial lines coincide.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Write the defining equation, retain exclusions, and then solve or compare.
Hint 2
Use the oriented determinant formula.
Worked solution

  1. Write the defining equation, retain exclusions, and then solve or compare.

  2. Apply the stated relation and retain its conditions.

    u×v=(0,0,20)u\times v=(0,0,20)

  3. Reversing the order changes the cross product sign.

The requested relation or conclusion is shown below.

u×v=(0,0,20)u\times v=(0,0,20)

Checks and common pitfalls: Reversing the order changes the cross product sign.

Reasoning checklist · self / teacher assessment
  • State a valid definition or model and its assumptions.
  • Show the intermediate mathematical relations, not only the final claim.
  • Check exclusions, units or the interpretation of the result.

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

10 / Standard#Your turn

Find the parallelogram area spanned by u,v.

u=(11,0,0),v=(0,2,0)u=(11,0,0),\quad v=(0,2,0)
  • Directions and normals must be nonzero; parallel directions do not prove that two spatial lines coincide.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Write the defining equation, retain exclusions, and then solve or compare.
Hint 2
Area=|u×v|
Worked solution

  1. Write the defining equation, retain exclusions, and then solve or compare.

  2. Apply the stated relation and retain its conditions.

    A=∣(0,0,22)∣=22A=|(0,0,22)|=22

  3. Magnitude removes orientation and gives nonnegative area.

The requested value is 22.

Checks and common pitfalls: Magnitude removes orientation and gives nonnegative area.

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

11 / Standard#Your turn

Classify the two spatial lines.

ℓ1=(s,0,0),ℓ2=(0,u,12)\ell_1=(s,0,0),\quad\ell_2=(0,u,12)
  • Directions and normals must be nonzero; parallel directions do not prove that two spatial lines coincide.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Write the defining equation, retain exclusions, and then solve or compare.
Hint 2
Equate all three coordinates.
Worked solution

  1. Write the defining equation, retain exclusions, and then solve or compare.

  2. Apply the stated relation and retain its conditions.

    s=0, u=0, 0=12 impossibles=0,\ u=0,\ 0=12\text{ impossible}

  3. Their directions are not parallel but their z-levels prevent intersection.

The requested relation or conclusion is shown below.

skew lines\text{skew lines}

Checks and common pitfalls: Their directions are not parallel but their z-levels prevent intersection.

Reasoning checklist · self / teacher assessment
  • State a valid definition or model and its assumptions.
  • Show the intermediate mathematical relations, not only the final claim.
  • Check exclusions, units or the interpretation of the result.

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

12 / Transfer#Your turn

Find the parallelogram area spanned by u,v.

u=(13,0,0),v=(0,2,0)u=(13,0,0),\quad v=(0,2,0)
  • Directions and normals must be nonzero; parallel directions do not prove that two spatial lines coincide.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Write the defining equation, retain exclusions, and then solve or compare.
Hint 2
Area=|u×v|
Worked solution

  1. Write the defining equation, retain exclusions, and then solve or compare.

  2. Apply the stated relation and retain its conditions.

    A=∣(0,0,26)∣=26A=|(0,0,26)|=26

  3. Magnitude removes orientation and gives nonnegative area.

The requested value is 26.

Checks and common pitfalls: Magnitude removes orientation and gives nonnegative area.

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

13 / Transfer#Your turn

Classify the two spatial lines.

ℓ1=(s,0,0),ℓ2=(0,u,14)\ell_1=(s,0,0),\quad\ell_2=(0,u,14)
  • Directions and normals must be nonzero; parallel directions do not prove that two spatial lines coincide.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Write the defining equation, retain exclusions, and then solve or compare.
Hint 2
Equate all three coordinates.
Worked solution

  1. Write the defining equation, retain exclusions, and then solve or compare.

  2. Apply the stated relation and retain its conditions.

    s=0, u=0, 0=14 impossibles=0,\ u=0,\ 0=14\text{ impossible}

  3. Their directions are not parallel but their z-levels prevent intersection.

The requested relation or conclusion is shown below.

skew lines\text{skew lines}

Checks and common pitfalls: Their directions are not parallel but their z-levels prevent intersection.

Reasoning checklist · self / teacher assessment
  • State a valid definition or model and its assumptions.
  • Show the intermediate mathematical relations, not only the final claim.
  • Check exclusions, units or the interpretation of the result.

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

    • Construct spatial equations and use normals, cross products and direction tests.
    • Which condition is essential in spatial lines, planes, spheres and cross products?
    • Can two nonparallel space lines have no intersection?

    Board plan

    • Model or definition: Construct spatial equations and use normals, cross products and direction tests.
      ℓ:P+su;Π:n⋅(X−P)=0\ell:P+su;\quad\Pi:n\cdot(X-P)=0
    • Conditions: Directions and normals must be nonzero; parallel directions do not prove that two spatial lines coincide.

    Anticipated thinking

    • Nonparallel spatial lines can be skew instead of intersecting.

    Assessment checklist

    • 1 mark: choose the correct representation and conditions.
    • 1 mark: establish the intermediate relation.
    • 1 mark: complete a connected calculation or proof.
    • 1 mark: interpret and check the conclusion.

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    Curriculum and source notes ↗