Kenneth ChengMathematical modeling
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Explain whether the zero vector has a unique direction.

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

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高一必修 第二册(A版).pdf · 6.1 · PDF 9 / printed page 2

Revisit first: Trigonometric identities and transformations

TOPIC 01

Concept of plane vectors

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

What you will be able to explain

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

Try it. Leave your reasoning visible.

Use one hint at a time. A correction explains what changed, not just the final answer.

01 / Standard#Your turn

Explain whether the zero vector has a unique direction.

0\mathbf0
  • Use the domain, units and sampling assumptions stated in the question.
Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1
Separate magnitude, direction and position; compare vectors by displacement.
Hint 2
Direction requires a nonzero displacement.
Worked solution

  1. Separate magnitude, direction and position; compare vectors by displacement.

  2. Calculate or simplify this relation.

    ∣0∣=0|\mathbf0|=0

  3. Conventions about parallelism do not create a physical direction for zero.

It has magnitude zero and no uniquely determined direction.

Checks and common pitfalls: Conventions about parallelism do not create a physical direction for zero.

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.

Focus on one question

Teacher preparation and assessment

Question sequence

  • What must be true before using the main rule for concept 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 concept of plane vectors.
  • Magnitude and direction: A nonzero vector has length and direction, independent of where it is drawn.
  • Zero and unit vectors: Zero has magnitude zero; a unit vector has magnitude one.
  • Close with: conditions → representation → reasoning → check.

Anticipated thinking

  • Expected reasoning: A nonzero vector has length and direction, independent of where it is drawn.
  • Expected reasoning: Zero has magnitude zero; a unit vector has magnitude one.
  • Expected correction: Equal lengths do not imply equal vectors.

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.

No sign-in. Work stays in this browser. Export before clearing browser data. Written reasoning is assessed with a checklist.

Curriculum and source notes ↗