Kenneth ChengMathematical modeling
← Senior Mathematics Studio

LEARN · EXPLAIN · REVISE

Applications of plane vectors

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

高一必修 第二册(A版).pdf · 6.4 · PDF 45 / printed page 38

Revisit first: Basis theorem and coordinate representation

TOPIC 01

Applications of plane vectors

Build understanding of applications of plane vectors through definitions, contrasting cases and justified applications.

What you will be able to explain

  • Connect geometric and algebraic forms of applications of plane vectors.
  • Connect representations and justify the steps, including boundary cases.
  • Explain a solution and apply the idea to a changed situation.

Geometry

Vectors yield metric relations, sine and cosine rules, and triangle centres.

Physical quantities

Add forces as vectors; calculate work with a dot product.

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 applications of plane vectors 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.

v=(2,1); |v|=2.2361; v·(2,1)=5.

v=(2,1); |v|=2.2361; v·(2,1)=5.

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

Find the third side using the cosine rule.

a=2,b=2,C=60∘a=2,\quad b=2,\quad C=60^{\circ}
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Translate the geometry or physical quantity into vectors and check angle conventions.
Hint 2
The given angle is the included angle between a and b.
Worked solution

  1. Translate the geometry or physical quantity into vectors and check angle conventions.

  2. Calculate or simplify this relation.

    c2=4+4−2(4)cos⁡60∘=4c^2=4+4-2(4)\cos60^{\circ}=4

  3. Take the positive square root for a side length.

The requested value is 2.

Checks and common pitfalls: Take the positive square root for a side length.

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

02 / Standard#Worked example

Find b using the sine rule.

a=2,A=30∘,B=90∘a=2,\quad A=30^{\circ},\quad B=90^{\circ}
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Translate the geometry or physical quantity into vectors and check angle conventions.
Hint 2
Match each side with its opposite angle.
Worked solution

  1. Translate the geometry or physical quantity into vectors and check angle conventions.

  2. Calculate or simplify this relation.

    b=asin⁡Bsin⁡A=211/2=4b=a\frac{\sin B}{\sin A}=2\frac1{1/2}=4

  3. The side opposite the right angle is the longest.

The requested value is 4.

Checks and common pitfalls: The side opposite the right angle is the longest.

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

03 / Transfer#Worked example

Explain why the sine rule in an SSA problem may give two triangles.

sin⁡B=s,0<s<1\sin B=s,\quad0<s<1
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Translate the geometry or physical quantity into vectors and check angle conventions.
Hint 2
The inverse sine calculator gives only a principal angle.
Worked solution

  1. Translate the geometry or physical quantity into vectors and check angle conventions.

  2. Calculate or simplify this relation.

    sin⁡B=sin⁡(180∘−B)\sin B=\sin(180^{\circ}-B)

  3. A supplementary candidate is discarded if the angle sum makes it impossible.

Both B and 180°−B have the same sine; each must be checked against the remaining angle and side conditions.

Checks and common pitfalls: A supplementary candidate is discarded if the angle sum makes it impossible.

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

Find the third side using the cosine rule.

a=3,b=3,C=60∘a=3,\quad b=3,\quad C=60^{\circ}
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Translate the geometry or physical quantity into vectors and check angle conventions.
Hint 2
The given angle is the included angle between a and b.
Worked solution

  1. Translate the geometry or physical quantity into vectors and check angle conventions.

  2. Calculate or simplify this relation.

    c2=9+9−2(9)cos⁡60∘=9c^2=9+9-2(9)\cos60^{\circ}=9

  3. Take the positive square root for a side length.

The requested value is 3.

Checks and common pitfalls: Take the positive square root for a side length.

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

05 / Foundation#Your turn

Find b using the sine rule.

a=3,A=30∘,B=90∘a=3,\quad A=30^{\circ},\quad B=90^{\circ}
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Translate the geometry or physical quantity into vectors and check angle conventions.
Hint 2
Match each side with its opposite angle.
Worked solution

  1. Translate the geometry or physical quantity into vectors and check angle conventions.

  2. Calculate or simplify this relation.

    b=asin⁡Bsin⁡A=311/2=6b=a\frac{\sin B}{\sin A}=3\frac1{1/2}=6

  3. The side opposite the right angle is the longest.

The requested value is 6.

Checks and common pitfalls: The side opposite the right angle is the longest.

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

06 / Foundation#Your turn

Find the triangle area.

a=6,b=3,C=30∘a=6,\quad b=3,\quad C=30^{\circ}
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Translate the geometry or physical quantity into vectors and check angle conventions.
Hint 2
Use half the product of two sides and sine of their included angle.
Worked solution

  1. Translate the geometry or physical quantity into vectors and check angle conventions.

  2. Calculate or simplify this relation.

    S=12absin⁡C=12(6)(3)(1/2)=4.5S=\frac12ab\sin C=\frac12(6)(3)(1/2)=4.5

  3. The angle must be the included angle, not an arbitrary triangle angle.

The requested value is 4.5.

Checks and common pitfalls: The angle must be the included angle, not an arbitrary triangle angle.

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

07 / Foundation#Your turn

Find the x-coordinate of the centroid.

A=(0,0),B=(9,0),C=(0,3)A=(0,0),\quad B=(9,0),\quad C=(0,3)
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Translate the geometry or physical quantity into vectors and check angle conventions.
Hint 2
Average the three vertex position vectors.
Worked solution

  1. Translate the geometry or physical quantity into vectors and check angle conventions.

  2. Calculate or simplify this relation.

    G=(A+B+C)/3=(3,1)G=(A+B+C)/3=(3,1)

  3. The centroid lies two-thirds along each median from the vertex.

The requested value is 3.

Checks and common pitfalls: The centroid lies two-thirds along each median from the vertex.

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

08 / Standard#Your turn

Find the work done by the constant force, in joules.

F=(3,2) N,s=(3,1) m\mathbf F=(3,2)\text{ N},\quad\mathbf s=(3,1)\text{ m}
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Translate the geometry or physical quantity into vectors and check angle conventions.
Hint 2
Work is the force component along displacement times distance.
Worked solution

  1. Translate the geometry or physical quantity into vectors and check angle conventions.

  2. Calculate or simplify this relation.

    W=F⋅s=3(3)+2=11W=\mathbf F\cdot\mathbf s=3(3)+2=11

  3. The dot product automatically accounts for the angle.

The requested value is 11.

Checks and common pitfalls: The dot product automatically accounts for the angle.

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

09 / Standard#Your turn

Find the magnitude of the resultant force.

F1=(9,0),F2=(0,12)\mathbf F_1=(9,0),\quad\mathbf F_2=(0,12)
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Translate the geometry or physical quantity into vectors and check angle conventions.
Hint 2
Add forces as vectors before taking magnitude.
Worked solution

  1. Translate the geometry or physical quantity into vectors and check angle conventions.

  2. Calculate or simplify this relation.

    ∣F1+F2∣=81+144=15|\mathbf F_1+\mathbf F_2|=\sqrt{81+144}=15

  3. Adding magnitudes directly is valid only for forces in the same direction.

The requested value is 15.

Checks and common pitfalls: Adding magnitudes directly is valid only for forces in the same direction.

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

10 / Standard#Your turn

Can side lengths 1,1,3 form a triangle to which the cosine rule applies? Explain.

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

Working and explanation

BUILD THE REASONING

Hint 1
Check geometric feasibility before applying formulas.
Hint 2
The two shorter sides must have sum exceeding the longest side.
Worked solution

  1. Check geometric feasibility before applying formulas.

  2. Calculate or simplify this relation.

    cos⁡C=1+1−92=−7/2∉[−1,1]\cos C=\frac{1+1-9}{2}=-7/2\notin[-1,1]

  3. An impossible cosine exposes inconsistent side data rather than a valid angle.

No: 1+1<3 violates the triangle inequality.

Checks and common pitfalls: An impossible cosine exposes inconsistent side data rather than a valid angle.

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

Find the third side using the cosine rule.

a=4,b=4,C=60∘a=4,\quad b=4,\quad C=60^{\circ}
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Translate the geometry or physical quantity into vectors and check angle conventions.
Hint 2
The given angle is the included angle between a and b.
Worked solution

  1. Translate the geometry or physical quantity into vectors and check angle conventions.

  2. Calculate or simplify this relation.

    c2=16+16−2(16)cos⁡60∘=16c^2=16+16-2(16)\cos60^{\circ}=16

  3. Take the positive square root for a side length.

The requested value is 4.

Checks and common pitfalls: Take the positive square root for a side length.

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

12 / Transfer#Your turn

Find b using the sine rule.

a=4,A=30∘,B=90∘a=4,\quad A=30^{\circ},\quad B=90^{\circ}
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Translate the geometry or physical quantity into vectors and check angle conventions.
Hint 2
Match each side with its opposite angle.
Worked solution

  1. Translate the geometry or physical quantity into vectors and check angle conventions.

  2. Calculate or simplify this relation.

    b=asin⁡Bsin⁡A=411/2=8b=a\frac{\sin B}{\sin A}=4\frac1{1/2}=8

  3. The side opposite the right angle is the longest.

The requested value is 8.

Checks and common pitfalls: The side opposite the right angle is the longest.

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

13 / Transfer#Your turn

Find the triangle area.

a=8,b=3,C=30∘a=8,\quad b=3,\quad C=30^{\circ}
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Translate the geometry or physical quantity into vectors and check angle conventions.
Hint 2
Use half the product of two sides and sine of their included angle.
Worked solution

  1. Translate the geometry or physical quantity into vectors and check angle conventions.

  2. Calculate or simplify this relation.

    S=12absin⁡C=12(8)(3)(1/2)=6S=\frac12ab\sin C=\frac12(8)(3)(1/2)=6

  3. The angle must be the included angle, not an arbitrary triangle angle.

The requested value is 6.

Checks and common pitfalls: The angle must be the included angle, not an arbitrary triangle 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 applications of plane vectors?
    • Which representation makes this task easier, and why?
    • Change one assumption. Does the conclusion survive?

    Board plan

    • Connect geometric and algebraic forms of applications of plane vectors.
    • Geometry: Vectors yield metric relations, sine and cosine rules, and triangle centres.
    • Physical quantities: Add forces as vectors; calculate work with a dot product.
    • Close with: conditions → representation → reasoning → check.

    Anticipated thinking

    • Expected reasoning: Vectors yield metric relations, sine and cosine rules, and triangle centres.
    • Expected reasoning: Add forces as vectors; calculate work with a dot product.
    • Expected correction: An SSA triangle may have two valid configurations.

    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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