Kenneth ChengMathematical modeling
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Basis theorem and coordinate representation

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

高一必修 第二册(A版).pdf · 6.3 · PDF 32 / printed page 25

Revisit first: Operations on plane vectors

TOPIC 01

Basis theorem and coordinate representation

Build understanding of basis theorem and coordinate representation through definitions, contrasting cases and justified applications.

What you will be able to explain

  • Connect geometric and algebraic forms of basis theorem and coordinate representation.
  • Connect representations and justify the steps, including boundary cases.
  • Explain a solution and apply the idea to a changed situation.

Basis

Two nonparallel vectors represent every plane vector uniquely.

Coordinate tests

Use scalar multiples for parallelism and zero dot product for perpendicular nonzero vectors.

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 basis theorem and coordinate representation 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 coefficient of e₁ in the given basis representation.

e1=(1,0),e2=(1,1),a=(5,3)\mathbf e_1=(1,0),\quad\mathbf e_2=(1,1),\quad\mathbf a=(5,3)
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Use a nonparallel basis or Cartesian coordinates and solve component equations.
Hint 2
Read the second coordinate first.
Worked solution

  1. Use a nonparallel basis or Cartesian coordinates and solve component equations.

  2. Calculate or simplify this relation.

    a=λ(1,0)+μ(1,1);μ=3, λ=2\mathbf a=\lambda(1,0)+\mu(1,1);\quad\mu=3,\ \lambda=2

  3. The two basis vectors are nonparallel, so the coefficients are unique.

The requested value is 2.

Checks and common pitfalls: The two basis vectors are nonparallel, so the coefficients are unique.

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

02 / Standard#Worked example

Find the midpoint x-coordinate.

A=(2,1),B=(6,5)A=(2,1),\quad B=(6,5)
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Use a nonparallel basis or Cartesian coordinates and solve component equations.
Hint 2
Average the corresponding endpoint coordinates.
Worked solution

  1. Use a nonparallel basis or Cartesian coordinates and solve component equations.

  2. Calculate or simplify this relation.

    M=A+B2=(4,3)M=\frac{A+B}{2}=(4,3)

  3. A midpoint is an affine combination with coefficients summing to one.

The requested value is 4.

Checks and common pitfalls: A midpoint is an affine combination with coefficients summing to one.

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

03 / Transfer#Worked example

Find the distance between the two points.

A=(2,2),B=(5,6)A=(2,2),\quad B=(5,6)
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Use a nonparallel basis or Cartesian coordinates and solve component equations.
Hint 2
Subtract coordinates before computing the norm.
Worked solution

  1. Use a nonparallel basis or Cartesian coordinates and solve component equations.

  2. Calculate or simplify this relation.

    ∣AB→∣=32+42=5|\overrightarrow{AB}|=\sqrt{3^2+4^2}=5

  3. Translation of both points does not change their distance.

The requested value is 5.

Checks and common pitfalls: Translation of both points does not change their distance.

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

04 / Foundation#Your turn

Find the coefficient of e₁ in the given basis representation.

e1=(1,0),e2=(1,1),a=(6,3)\mathbf e_1=(1,0),\quad\mathbf e_2=(1,1),\quad\mathbf a=(6,3)
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Use a nonparallel basis or Cartesian coordinates and solve component equations.
Hint 2
Read the second coordinate first.
Worked solution

  1. Use a nonparallel basis or Cartesian coordinates and solve component equations.

  2. Calculate or simplify this relation.

    a=λ(1,0)+μ(1,1);μ=3, λ=3\mathbf a=\lambda(1,0)+\mu(1,1);\quad\mu=3,\ \lambda=3

  3. The two basis vectors are nonparallel, so the coefficients are unique.

The requested value is 3.

Checks and common pitfalls: The two basis vectors are nonparallel, so the coefficients are unique.

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

05 / Foundation#Your turn

Find the midpoint x-coordinate.

A=(3,1),B=(7,5)A=(3,1),\quad B=(7,5)
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Use a nonparallel basis or Cartesian coordinates and solve component equations.
Hint 2
Average the corresponding endpoint coordinates.
Worked solution

  1. Use a nonparallel basis or Cartesian coordinates and solve component equations.

  2. Calculate or simplify this relation.

    M=A+B2=(5,3)M=\frac{A+B}{2}=(5,3)

  3. A midpoint is an affine combination with coefficients summing to one.

The requested value is 5.

Checks and common pitfalls: A midpoint is an affine combination with coefficients summing to one.

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

06 / Foundation#Your turn

Find m for the two vectors to be parallel.

a=(2,3),b=(6,m)\mathbf a=(2,3),\quad\mathbf b=(6,m)
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Use a nonparallel basis or Cartesian coordinates and solve component equations.
Hint 2
Compare the first coordinate to find the scalar multiple.
Worked solution

  1. Use a nonparallel basis or Cartesian coordinates and solve component equations.

  2. Calculate or simplify this relation.

    b=3a⇒m=9\mathbf b=3\mathbf a\Rightarrow m=9

  3. The second coordinate must use the same multiplier.

The requested value is 9.

Checks and common pitfalls: The second coordinate must use the same multiplier.

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

07 / Foundation#Your turn

Find m for the vectors to be perpendicular.

a=(1,3),b=(m,2)\mathbf a=(1,3),\quad\mathbf b=(m,2)
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Use a nonparallel basis or Cartesian coordinates and solve component equations.
Hint 2
Set the dot product equal to zero.
Worked solution

  1. Use a nonparallel basis or Cartesian coordinates and solve component equations.

  2. Calculate or simplify this relation.

    m+2(3)=0⇒m=−6m+2(3)=0\Rightarrow m=-6

  3. Both vectors remain nonzero for the stated coordinates.

The requested value is -6.

Checks and common pitfalls: Both vectors remain nonzero for the stated coordinates.

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

08 / Standard#Your turn

Explain why parallel vectors cannot form a basis for the plane.

e2=2e1\mathbf e_2=2\mathbf e_1
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Use a nonparallel basis or Cartesian coordinates and solve component equations.
Hint 2
Substitute the dependence relation.
Worked solution

  1. Use a nonparallel basis or Cartesian coordinates and solve component equations.

  2. Calculate or simplify this relation.

    ae1+be2=(a+2b)e1a\mathbf e_1+b\mathbf e_2=(a+2b)\mathbf e_1

  3. They cannot represent a vector outside that line.

All their linear combinations stay on one line.

Checks and common pitfalls: They cannot represent a vector outside that line.

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 signed scalar projection on the positive x-axis.

a=(−3,3)\mathbf a=(-3,3)
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Use a nonparallel basis or Cartesian coordinates and solve component equations.
Hint 2
Dot with a unit vector along the target direction.
Worked solution

  1. Use a nonparallel basis or Cartesian coordinates and solve component equations.

  2. Calculate or simplify this relation.

    a⋅(1,0)=−3\mathbf a\cdot(1,0)=-3

  3. A signed projection may be negative even though a length is nonnegative.

The requested value is -3.

Checks and common pitfalls: A signed projection may be negative even though a length is nonnegative.

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

10 / Standard#Your turn

Find the distance between the two points.

A=(3,2),B=(6,6)A=(3,2),\quad B=(6,6)
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Use a nonparallel basis or Cartesian coordinates and solve component equations.
Hint 2
Subtract coordinates before computing the norm.
Worked solution

  1. Use a nonparallel basis or Cartesian coordinates and solve component equations.

  2. Calculate or simplify this relation.

    ∣AB→∣=32+42=5|\overrightarrow{AB}|=\sqrt{3^2+4^2}=5

  3. Translation of both points does not change their distance.

The requested value is 5.

Checks and common pitfalls: Translation of both points does not change their distance.

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

11 / Standard#Your turn

Find the coefficient of e₁ in the given basis representation.

e1=(1,0),e2=(1,1),a=(7,3)\mathbf e_1=(1,0),\quad\mathbf e_2=(1,1),\quad\mathbf a=(7,3)
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Use a nonparallel basis or Cartesian coordinates and solve component equations.
Hint 2
Read the second coordinate first.
Worked solution

  1. Use a nonparallel basis or Cartesian coordinates and solve component equations.

  2. Calculate or simplify this relation.

    a=λ(1,0)+μ(1,1);μ=3, λ=4\mathbf a=\lambda(1,0)+\mu(1,1);\quad\mu=3,\ \lambda=4

  3. The two basis vectors are nonparallel, so the coefficients are unique.

The requested value is 4.

Checks and common pitfalls: The two basis vectors are nonparallel, so the coefficients are unique.

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

12 / Transfer#Your turn

Find the midpoint x-coordinate.

A=(4,1),B=(8,5)A=(4,1),\quad B=(8,5)
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Use a nonparallel basis or Cartesian coordinates and solve component equations.
Hint 2
Average the corresponding endpoint coordinates.
Worked solution

  1. Use a nonparallel basis or Cartesian coordinates and solve component equations.

  2. Calculate or simplify this relation.

    M=A+B2=(6,3)M=\frac{A+B}{2}=(6,3)

  3. A midpoint is an affine combination with coefficients summing to one.

The requested value is 6.

Checks and common pitfalls: A midpoint is an affine combination with coefficients summing to one.

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

13 / Transfer#Your turn

Find m for the two vectors to be parallel.

a=(2,3),b=(8,m)\mathbf a=(2,3),\quad\mathbf b=(8,m)
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Use a nonparallel basis or Cartesian coordinates and solve component equations.
Hint 2
Compare the first coordinate to find the scalar multiple.
Worked solution

  1. Use a nonparallel basis or Cartesian coordinates and solve component equations.

  2. Calculate or simplify this relation.

    b=4a⇒m=12\mathbf b=4\mathbf a\Rightarrow m=12

  3. The second coordinate must use the same multiplier.

The requested value is 12.

Checks and common pitfalls: The second coordinate must use the same multiplier.

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 basis theorem and coordinate representation?
    • Which representation makes this task easier, and why?
    • Change one assumption. Does the conclusion survive?

    Board plan

    • Connect geometric and algebraic forms of basis theorem and coordinate representation.
    • Basis: Two nonparallel vectors represent every plane vector uniquely.
    • Coordinate tests: Use scalar multiples for parallelism and zero dot product for perpendicular nonzero vectors.
    • Close with: conditions → representation → reasoning → check.

    Anticipated thinking

    • Expected reasoning: Two nonparallel vectors represent every plane vector uniquely.
    • Expected reasoning: Use scalar multiples for parallelism and zero dot product for perpendicular nonzero vectors.
    • Expected correction: Parallel basis candidates do not span the whole 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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