Kenneth ChengMathematical modeling
← Senior Mathematics Studio

LEARN · EXPLAIN · REVISE

Find the distance from P to the plane.

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 / Foundation#Your turn

Find the distance from P to the plane.

P=(0,0,23),Π:z=2P=(0,0,23),\quad \Pi:z=2
  • Use acute line/plane angles; normals must be nonzero and distances nonnegative.
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.
d=∣zP−2∣d=|z_P-2|
Worked solution

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

  2. Apply the stated relation and retain its conditions.

    d=∣23−2∣=21d=|23-2|=21

  3. The normal displacement gives the shortest distance.

The requested value is 21.

Checks and common pitfalls: The normal displacement gives the shortest distance.

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 ↗