Kenneth ChengMathematical modeling
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Applications of space vectors

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

高二選擇性必修 第一册(A版).pdf · 1.4 · PDF 31 / printed page 26

Revisit first: Coordinate representation of space vectors

TOPIC 01

Applications of space vectors

Use direction and normal vectors to establish angles, distances and spatial relations.

What you will be able to explain

  • Use direction and normal vectors to establish angles, distances and spatial relations.
  • Justify the method and check the conditions in a new situation.

Normals

A plane normal is perpendicular to every direction in the plane.

n⋅(X−P)=0n\cdot(X-P)=0

Angles and distance

Use acute line/plane angles; normals must be nonzero and distances nonnegative.

d=∣n⋅(P−P0)∣∣n∣d=\frac{|n\cdot(P-P_0)|}{|n|}

PREDICT → EXPLORE → EXPLAIN → TRANSFER

Make a prediction, then explore the relationship.

Lesson question: Predict whether translating a plane changes its direction or its distance from a fixed point.

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: Calculate two valid cases and explain the change using the defining relation.

Transfer: Compare parallel planes with the same normal but different constants.

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 cosine of the acute angle between the lines.

u=(1,0,0),v=(1,2,0)u=(1,0,0), v=(1,2,0)
  • Use acute line/plane angles; normals must be nonzero and distances nonnegative.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Use the dot product to connect coordinates with length or angle.
Hint 2
Use this intermediate relation.
cosθ=∣u⋅v∣/(∣u∣∣v∣)cosθ=|u·v|/(|u||v|)
Worked solution

  1. Use the dot product to connect coordinates with length or angle.

  2. Apply the stated relation and retain its conditions.

    cos⁡θ=1/5\cos\theta=1/\sqrt{5}

  3. The absolute dot product selects the acute line angle.

The requested value is 0.4472135955.

Checks and common pitfalls: The absolute dot product selects the acute line angle.

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

02 / Standard#Worked example

Find the distance from P to the plane.

P=(3,0,0),Π:3x+4y=0P=(3,0,0),\quad \Pi:3x+4y=0
  • Use acute line/plane angles; normals must be nonzero and distances nonnegative.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Use the dot product to connect coordinates with length or angle.
Hint 2
Use this intermediate relation.
d=∣3xP+4yP∣/5d=|3x_P+4y_P|/5
Worked solution

  1. Use the dot product to connect coordinates with length or angle.

  2. Apply the stated relation and retain its conditions.

    d=3(3)/32+42=1.8d=3(3)/\sqrt{3^2+4^2}=1.8

  3. Normalize the plane normal; raw substitution is not a distance.

The requested value is 1.8.

Checks and common pitfalls: Normalize the plane normal; raw substitution is not a distance.

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

03 / Transfer#Worked example

Find the plane through P with normal n.

P=(1,2,4),n=(2,−1,3)P=(1,2,4),\quad n=(2,-1,3)
  • Use acute line/plane angles; normals must be nonzero and distances nonnegative.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Translate the geometric condition into a coordinate equation.
Hint 2
Use this intermediate relation.
n⋅(X−P)=0n\cdot(X-P)=0
Worked solution

  1. Translate the geometric condition into a coordinate equation.

  2. Apply the stated relation and retain its conditions.

    2(x−1)−(y−2)+3(z−4)=02(x-1)-(y-2)+3(z-4)=0

  3. Apply the stated relation and retain its conditions.

    2x−y+3z=122x-y+3z=12

  4. The constant depends on the point; the coefficients specify the normal.

The requested relation or conclusion is shown below.

2x−y+3z=122x-y+3z=12

Checks and common pitfalls: The constant depends on the point; the coefficients specify the normal.

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 cosine of the acute angle between the lines.

u=(1,0,0),v=(1,5,0)u=(1,0,0), v=(1,5,0)
  • Use acute line/plane angles; normals must be nonzero and distances nonnegative.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Use the dot product to connect coordinates with length or angle.
Hint 2
Use this intermediate relation.
cosθ=∣u⋅v∣/(∣u∣∣v∣)cosθ=|u·v|/(|u||v|)
Worked solution

  1. Use the dot product to connect coordinates with length or angle.

  2. Apply the stated relation and retain its conditions.

    cos⁡θ=1/26\cos\theta=1/\sqrt{26}

  3. The absolute dot product selects the acute line angle.

The requested value is 0.196116135138.

Checks and common pitfalls: The absolute dot product selects the acute line angle.

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

05 / Foundation#Your turn

Find sin α for the angle of the line with the plane.

u=(1,0,1),Π:z=0u=(1,0,1),\quad \Pi:z=0
  • Use acute line/plane angles; normals must be nonzero and distances nonnegative.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Use the dot product to connect coordinates with length or angle.
Hint 2
Use this intermediate relation.
sin⁡α=∣u⋅n∣/(∣u∣∣n∣)\sin\alpha=|u\cdot n|/(|u||n|)
Worked solution

  1. Use the dot product to connect coordinates with length or angle.

  2. Apply the stated relation and retain its conditions.

    n=(0,0,1)n=(0,0,1)

  3. Apply the stated relation and retain its conditions.

    sin⁡α=1/2\sin\alpha=1/\sqrt2

  4. Use sine for the line-plane angle because the normal angle is its complement.

The requested value is 0.707106781187.

Checks and common pitfalls: Use sine for the line-plane angle because the normal angle is its complement.

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

06 / Foundation#Your turn

Find the cosine of the acute angle between the planes.

n1=(1,1,0),n2=(0,1,1)n_1=(1,1,0),\quad n_2=(0,1,1)
  • Use acute line/plane angles; normals must be nonzero and distances nonnegative.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Use the dot product to connect coordinates with length or angle.
Hint 2
Use this intermediate relation.
cos⁡θ=∣n1⋅n2∣/(∣n1∣∣n2∣)\cos\theta=|n_1\cdot n_2|/(|n_1||n_2|)
Worked solution

  1. Use the dot product to connect coordinates with length or angle.

  2. Apply the stated relation and retain its conditions.

    n1⋅n2=1n_1\cdot n_2=1

  3. Apply the stated relation and retain its conditions.

    cos⁡θ=1/2\cos\theta=1/2

  4. The angle between normals determines the acute plane angle.

The requested value is 0.5.

Checks and common pitfalls: The angle between normals determines the acute plane angle.

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

07 / Foundation#Your turn

Find the distance from P to the plane.

P=(0,0,8),Π:z=2P=(0,0,8),\quad \Pi:z=2
  • Use acute line/plane angles; normals must be nonzero and distances nonnegative.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Use the dot product to connect coordinates with length or angle.
Hint 2
Use this intermediate relation.
d=∣zP−2∣d=|z_P-2|
Worked solution

  1. Use the dot product to connect coordinates with length or angle.

  2. Apply the stated relation and retain its conditions.

    d=∣8−2∣=6d=|8-2|=6

  3. The normal displacement gives the shortest distance.

The requested value is 6.

Checks and common pitfalls: The normal displacement gives the shortest distance.

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

08 / Standard#Your turn

Find the distance from P to the plane.

P=(0,0,9),Π:z=2P=(0,0,9),\quad \Pi:z=2
  • Use acute line/plane angles; normals must be nonzero and distances nonnegative.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Use the dot product to connect coordinates with length or angle.
Hint 2
Use this intermediate relation.
d=∣zP−2∣d=|z_P-2|
Worked solution

  1. Use the dot product to connect coordinates with length or angle.

  2. Apply the stated relation and retain its conditions.

    d=∣9−2∣=7d=|9-2|=7

  3. The normal displacement gives the shortest distance.

The requested value is 7.

Checks and common pitfalls: The normal displacement gives the shortest distance.

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

09 / Standard#Your turn

Find the distance from P to the plane.

P=(10,0,0),Π:3x+4y=0P=(10,0,0),\quad \Pi:3x+4y=0
  • Use acute line/plane angles; normals must be nonzero and distances nonnegative.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Use the dot product to connect coordinates with length or angle.
Hint 2
Use this intermediate relation.
d=∣3xP+4yP∣/5d=|3x_P+4y_P|/5
Worked solution

  1. Use the dot product to connect coordinates with length or angle.

  2. Apply the stated relation and retain its conditions.

    d=3(10)/32+42=6d=3(10)/\sqrt{3^2+4^2}=6

  3. Normalize the plane normal; raw substitution is not a distance.

The requested value is 6.

Checks and common pitfalls: Normalize the plane normal; raw substitution is not a distance.

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

10 / Standard#Your turn

Does the entire line lie in the plane?

X=(0,0,11)+s(1,−1,0),Π:x+y+z=11X=(0,0,11)+s(1,-1,0), \Pi:x+y+z=11
  • Use acute line/plane angles; normals must be nonzero and distances nonnegative.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Use the dot product to connect coordinates with length or angle.
Hint 2
n·u=0 and the initial point is in the plane.
Worked solution

  1. Use the dot product to connect coordinates with length or angle.

  2. Apply the stated relation and retain its conditions.

    x+y+z=s−s+11=11x+y+z=s-s+11=11

  3. Direction parallelism plus an incident point proves containment.

The requested relation or conclusion is shown below.

ℓ⊂Π\ell\subset\Pi

Checks and common pitfalls: Direction parallelism plus an incident point proves containment.

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.

11 / Standard#Your turn

Find the plane through P with normal n.

P=(1,2,12),n=(2,−1,3)P=(1,2,12),\quad n=(2,-1,3)
  • Use acute line/plane angles; normals must be nonzero and distances nonnegative.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Translate the geometric condition into a coordinate equation.
Hint 2
Use this intermediate relation.
n⋅(X−P)=0n\cdot(X-P)=0
Worked solution

  1. Translate the geometric condition into a coordinate equation.

  2. Apply the stated relation and retain its conditions.

    2(x−1)−(y−2)+3(z−12)=02(x-1)-(y-2)+3(z-12)=0

  3. Apply the stated relation and retain its conditions.

    2x−y+3z=362x-y+3z=36

  4. The constant depends on the point; the coefficients specify the normal.

The requested relation or conclusion is shown below.

2x−y+3z=362x-y+3z=36

Checks and common pitfalls: The constant depends on the point; the coefficients specify the normal.

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

Does the entire line lie in the plane?

X=(0,0,13)+s(1,−1,0),Π:x+y+z=13X=(0,0,13)+s(1,-1,0), \Pi:x+y+z=13
  • Use acute line/plane angles; normals must be nonzero and distances nonnegative.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Use the dot product to connect coordinates with length or angle.
Hint 2
n·u=0 and the initial point is in the plane.
Worked solution

  1. Use the dot product to connect coordinates with length or angle.

  2. Apply the stated relation and retain its conditions.

    x+y+z=s−s+13=13x+y+z=s-s+13=13

  3. Direction parallelism plus an incident point proves containment.

The requested relation or conclusion is shown below.

ℓ⊂Π\ell\subset\Pi

Checks and common pitfalls: Direction parallelism plus an incident point proves containment.

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.

13 / Transfer#Your turn

Find the plane through P with normal n.

P=(1,2,14),n=(2,−1,3)P=(1,2,14),\quad n=(2,-1,3)
  • Use acute line/plane angles; normals must be nonzero and distances nonnegative.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Translate the geometric condition into a coordinate equation.
Hint 2
Use this intermediate relation.
n⋅(X−P)=0n\cdot(X-P)=0
Worked solution

  1. Translate the geometric condition into a coordinate equation.

  2. Apply the stated relation and retain its conditions.

    2(x−1)−(y−2)+3(z−14)=02(x-1)-(y-2)+3(z-14)=0

  3. Apply the stated relation and retain its conditions.

    2x−y+3z=422x-y+3z=42

  4. The constant depends on the point; the coefficients specify the normal.

The requested relation or conclusion is shown below.

2x−y+3z=422x-y+3z=42

Checks and common pitfalls: The constant depends on the point; the coefficients specify the normal.

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

    • Use direction and normal vectors to establish angles, distances and spatial relations.
    • Which condition is essential in applications of space vectors?
    • Predict whether translating a plane changes its direction or its distance from a fixed point.

    Board plan

    • Normals: A plane normal is perpendicular to every direction in the plane.
      n⋅(X−P)=0n\cdot(X-P)=0
    • Angles and distance: Use acute line/plane angles; normals must be nonzero and distances nonnegative.
      d=∣n⋅(P−P0)∣∣n∣d=\frac{|n\cdot(P-P_0)|}{|n|}

    Anticipated thinking

    • The line-plane angle is complementary to the angle between the line direction and plane normal.

    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 ↗