Kenneth ChengMathematical modeling
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LEARN · EXPLAIN · REVISE

Space vectors and their operations

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

高二選擇性必修 第一册(A版).pdf · 1.1 · PDF 7 / printed page 2

Revisit first: Operations on plane vectors

TOPIC 01

Space vectors and their operations

Add, scale and multiply space vectors; distinguish scalar and vector results.

What you will be able to explain

  • Add, scale and multiply space vectors; distinguish scalar and vector results.
  • Justify the method and check the conditions in a new situation.

Operations

Addition and scaling act component by component.

a+b=(a1+b1,a2+b2,a3+b3)a+b=(a_1+b_1,a_2+b_2,a_3+b_3)

Angle conditions

The angle formula requires two nonzero vectors; a dot product is a scalar.

cos⁡θ=a⋅b∣a∣∣b∣\cos\theta=\frac{a\cdot b}{|a||b|}

PREDICT → EXPLORE → EXPLAIN → TRANSFER

Make a prediction, then explore the relationship.

Lesson question: Predict which changes affect length but preserve direction.

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 multiplication by a positive, negative and zero scalar.

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 first component of a+b.

a=(2,2,−1),b=(3,−2,4)a=(2,2,-1),\quad b=(3,-2,4)
  • The angle formula requires two nonzero vectors; a dot product is a scalar.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Use vector components and the distributive laws.
Hint 2
Use this intermediate relation.
(a+b)1=a1+b1(a+b)_1=a_1+b_1
Worked solution

  1. Use vector components and the distributive laws.

  2. Apply the stated relation and retain its conditions.

    (a+b)=(5,0,3)(a+b)=(5,0,3)

  3. Cancellation of other components does not affect the first component.

The requested value is 5.

Checks and common pitfalls: Cancellation of other components does not affect the first component.

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

02 / Standard#Worked example

Find cos θ between a and b.

a=(3,0,0),b=(3,3,0)a=(3,0,0),\quad b=(3,3,0)
  • The angle formula requires two nonzero vectors; a dot product is a scalar.
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=(a⋅b)/(∣a∣∣b∣)\cos\theta=(a\cdot b)/(|a||b|)
Worked solution

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

  2. Apply the stated relation and retain its conditions.

    a⋅b=9,∣a∣=3,∣b∣=32a\cdot b=9,\quad |a|=3,\quad |b|=3\sqrt2

  3. Apply the stated relation and retain its conditions.

    cos⁡θ=1/2\cos\theta=1/\sqrt2

  4. A common positive scale cancels in the angle formula.

The requested value is 0.707106781187.

Checks and common pitfalls: A common positive scale cancels in the angle formula.

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

03 / Transfer#Worked example

Three mutually perpendicular edges have lengths t,3,4. Find the squared space diagonal.

t=4t=4
  • The angle formula requires two nonzero vectors; a dot product is a scalar.
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.
∣u+v+w∣2=∣u∣2+∣v∣2+∣w∣2|u+v+w|^2=|u|^2+|v|^2+|w|^2
Worked solution

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

  2. Apply the stated relation and retain its conditions.

    u⋅v=u⋅w=v⋅w=0u\cdot v=u\cdot w=v\cdot w=0

  3. Apply the stated relation and retain its conditions.

    d2=42+32+42=41d^2=4^2+3^2+4^2=41

  4. Orthogonality removes all cross terms.

The requested value is 41.

Checks and common pitfalls: Orthogonality removes all cross terms.

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

04 / Foundation#Your turn

Find the first component of a+b.

a=(5,2,−1),b=(3,−2,4)a=(5,2,-1),\quad b=(3,-2,4)
  • The angle formula requires two nonzero vectors; a dot product is a scalar.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Use vector components and the distributive laws.
Hint 2
Use this intermediate relation.
(a+b)1=a1+b1(a+b)_1=a_1+b_1
Worked solution

  1. Use vector components and the distributive laws.

  2. Apply the stated relation and retain its conditions.

    (a+b)=(8,0,3)(a+b)=(8,0,3)

  3. Cancellation of other components does not affect the first component.

The requested value is 8.

Checks and common pitfalls: Cancellation of other components does not affect the first component.

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

05 / Foundation#Your turn

Find the third component of 2a−3b.

a=(1,6,−2),b=(2,0,6)a=(1,6,-2),\quad b=(2,0,6)
  • The angle formula requires two nonzero vectors; a dot product is a scalar.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Use vector components and the distributive laws.
Hint 2
Use this intermediate relation.
(2a−3b)3=2a3−3b3(2a-3b)_3=2a_3-3b_3
Worked solution

  1. Use vector components and the distributive laws.

  2. Apply the stated relation and retain its conditions.

    2(−2)−3(6)=−222(-2)-3(6)=-22

  3. Scale each entire vector before subtracting.

The requested value is -22.

Checks and common pitfalls: Scale each entire vector before subtracting.

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

06 / Foundation#Your turn

Find the squared length of a.

a=(7,2,3)a=(7,2,3)
  • The angle formula requires two nonzero vectors; a dot product is a scalar.
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.
∣a∣2=a⋅a|a|^2=a\cdot a
Worked solution

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

  2. Apply the stated relation and retain its conditions.

    ∣a∣2=72+22+32=62|a|^2=7^2+2^2+3^2=62

  3. Length squared is the sum of component squares.

The requested value is 62.

Checks and common pitfalls: Length squared is the sum of component squares.

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

07 / Foundation#Your turn

Calculate a·b.

a=(8,1,2),b=(2,−3,1)a=(8,1,2),\quad b=(2,-3,1)
  • The angle formula requires two nonzero vectors; a dot product is a scalar.
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.
a⋅b=a1b1+a2b2+a3b3a\cdot b=a_1b_1+a_2b_2+a_3b_3
Worked solution

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

  2. Apply the stated relation and retain its conditions.

    a⋅b=2(8)−3+2=15a\cdot b=2(8)-3+2=15

  3. A dot product is a signed scalar, not a vector.

The requested value is 15.

Checks and common pitfalls: A dot product is a signed scalar, not a vector.

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

08 / Standard#Your turn

Calculate a·b.

a=(9,1,2),b=(2,−3,1)a=(9,1,2),\quad b=(2,-3,1)
  • The angle formula requires two nonzero vectors; a dot product is a scalar.
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.
a⋅b=a1b1+a2b2+a3b3a\cdot b=a_1b_1+a_2b_2+a_3b_3
Worked solution

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

  2. Apply the stated relation and retain its conditions.

    a⋅b=2(9)−3+2=17a\cdot b=2(9)-3+2=17

  3. A dot product is a signed scalar, not a vector.

The requested value is 17.

Checks and common pitfalls: A dot product is a signed scalar, not a vector.

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

09 / Standard#Your turn

Find cos θ between a and b.

a=(10,0,0),b=(10,10,0)a=(10,0,0),\quad b=(10,10,0)
  • The angle formula requires two nonzero vectors; a dot product is a scalar.
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=(a⋅b)/(∣a∣∣b∣)\cos\theta=(a\cdot b)/(|a||b|)
Worked solution

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

  2. Apply the stated relation and retain its conditions.

    a⋅b=100,∣a∣=10,∣b∣=102a\cdot b=100,\quad |a|=10,\quad |b|=10\sqrt2

  3. Apply the stated relation and retain its conditions.

    cos⁡θ=1/2\cos\theta=1/\sqrt2

  4. A common positive scale cancels in the angle formula.

The requested value is 0.707106781187.

Checks and common pitfalls: A common positive scale cancels in the angle formula.

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

10 / Standard#Your turn

Are a and b parallel? Justify your answer.

a=(1,2,11),b=(2,4,22)a=(1,2,11),\quad b=(2,4,22)
  • The angle formula requires two nonzero vectors; a dot product is a scalar.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Use vector components and the distributive laws.
Hint 2
Use this intermediate relation.
b=λab=\lambda a
Worked solution

  1. Use vector components and the distributive laws.

  2. Apply the stated relation and retain its conditions.

    (2,4,22)=2(1,2,11)(2,4,22)=2(1,2,11)

  3. All components have the same scalar multiplier.

The requested relation or conclusion is shown below.

b=2ab=2a

Checks and common pitfalls: All components have the same scalar multiplier.

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

Three mutually perpendicular edges have lengths t,3,4. Find the squared space diagonal.

t=12t=12
  • The angle formula requires two nonzero vectors; a dot product is a scalar.
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.
∣u+v+w∣2=∣u∣2+∣v∣2+∣w∣2|u+v+w|^2=|u|^2+|v|^2+|w|^2
Worked solution

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

  2. Apply the stated relation and retain its conditions.

    u⋅v=u⋅w=v⋅w=0u\cdot v=u\cdot w=v\cdot w=0

  3. Apply the stated relation and retain its conditions.

    d2=122+32+42=169d^2=12^2+3^2+4^2=169

  4. Orthogonality removes all cross terms.

The requested value is 169.

Checks and common pitfalls: Orthogonality removes all cross terms.

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

12 / Transfer#Your turn

Are a and b parallel? Justify your answer.

a=(1,2,13),b=(2,4,26)a=(1,2,13),\quad b=(2,4,26)
  • The angle formula requires two nonzero vectors; a dot product is a scalar.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Use vector components and the distributive laws.
Hint 2
Use this intermediate relation.
b=λab=\lambda a
Worked solution

  1. Use vector components and the distributive laws.

  2. Apply the stated relation and retain its conditions.

    (2,4,26)=2(1,2,13)(2,4,26)=2(1,2,13)

  3. All components have the same scalar multiplier.

The requested relation or conclusion is shown below.

b=2ab=2a

Checks and common pitfalls: All components have the same scalar multiplier.

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

Three mutually perpendicular edges have lengths t,3,4. Find the squared space diagonal.

t=14t=14
  • The angle formula requires two nonzero vectors; a dot product is a scalar.
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.
∣u+v+w∣2=∣u∣2+∣v∣2+∣w∣2|u+v+w|^2=|u|^2+|v|^2+|w|^2
Worked solution

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

  2. Apply the stated relation and retain its conditions.

    u⋅v=u⋅w=v⋅w=0u\cdot v=u\cdot w=v\cdot w=0

  3. Apply the stated relation and retain its conditions.

    d2=142+32+42=221d^2=14^2+3^2+4^2=221

  4. Orthogonality removes all cross terms.

The requested value is 221.

Checks and common pitfalls: Orthogonality removes all cross terms.

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

Focus on one question

End-of-lesson check and correction

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    Choose a foundation skill to revisit ↗

    Teacher preparation and assessment

    Question sequence

    • Add, scale and multiply space vectors; distinguish scalar and vector results.
    • Which condition is essential in space vectors and their operations?
    • Predict which changes affect length but preserve direction.

    Board plan

    • Operations: Addition and scaling act component by component.
      a+b=(a1+b1,a2+b2,a3+b3)a+b=(a_1+b_1,a_2+b_2,a_3+b_3)
    • Angle conditions: The angle formula requires two nonzero vectors; a dot product is a scalar.
      cos⁡θ=a⋅b∣a∣∣b∣\cos\theta=\frac{a\cdot b}{|a||b|}

    Anticipated thinking

    • A zero dot product indicates perpendicular directions only when both vectors are nonzero.

    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 ↗