Kenneth ChengMathematical modeling
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Are a and b parallel? Justify your answer.

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

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TOPIC 01

Space vectors and solid geometry: mixed review

A mixed assessment: identify the method, justify it and revise your reasoning.

What you will be able to explain

  • Connect the chapter skills without relying on the order of the exercises.

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

Are a and b parallel? Justify your answer.

a=(1,2,28),b=(2,4,56)a=(1,2,28),\quad b=(2,4,56)
  • 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.

    (2,4,56)=2(1,2,28)(2,4,56)=2(1,2,28)

  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.

Focus on one question

Teacher preparation and assessment

Question sequence

  • Ask students to name the relevant condition before calculating.

Board plan

  • Compare valid methods and annotate their conditions.

Anticipated thinking

  • A correct final value may still hide a missing assumption.

Assessment checklist

  • Check the method, conditions, reasoning and interpretation separately.

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

Curriculum and source notes ↗