Learn with Amar
Teaching video
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Undergraduate chapters and video availability01 · Read and understand
What you will learn
- Project a displacement onto the unit normal of a plane.
- Justify the conclusion "Distance=|6|/3=2" using the stated assumptions.
Before you start
Dot products and vector norms.
Keep paper nearby. Read the question once for the context, then again to identify what is known and what you need to find.
Start with a question
Find the distance from P=(1,2,3) to the plane x+2y+2z=5.
Why this math matters
Project a displacement onto the unit normal of a plane. This worked micro-lesson connects a precise mathematical condition to a conclusion you can check. The transfer task asks you to change the setting and decide which parts of the reasoning still apply.
Set up the model
A useful answer starts with clear assumptions:
- The plane normal is nonzero.
- Distances are Euclidean.
02 · Work through the example
Follow the reasoning, one step at a time.
Try to predict the next step before reading it. After each calculation, explain why the operation makes sense and how it helps answer the original question.
Work through it with Amar
See this example unfold.
The complete worked example, one idea at a time.
Measure perpendicular distance to a plane
PausedQuestion: Start with the question. Paused.
Question
Start with the question
Find the distance from P=(1,2,3) to the plane x+2y+2z=5.
Before you calculate
Read what is known and what you need to find. Make a prediction before moving to the first calculation.
Starts paused. Play advances through the full text at a reading pace; pause whenever you need more time. Previous, Next, and the phase buttons let you set your own pace. Playback pauses when this walkthrough leaves the screen or you switch tabs.
Your device’s reduced-motion setting keeps each phase still. Manual controls remain available. The full written solution stays below.
Build the model
n=(1,2,2), |n|=3
The coefficients supply a normal, whose length must be normalized.
Work through the mathematics
n·P−5=1+4+6−5=6
This is a signed normal-coordinate difference before normalization.
Check the conclusion
Distance=|6|/3=2
Dividing by the normal length converts the equation residual into geometric distance.
The result
Distance=|6|/3=2
Dividing by the normal length converts the equation residual into geometric distance.
Common mistakes to catch
- The equation residual alone is not a distance.
- Keep the absolute value when reporting unsigned distance.
03 · Practice independently
Try it before revealing the answer.
Use paper or a calculator as needed. Write your units and reasoning, then open the hint or explanation to check your approach.
Practice 1
Find the closest point on the plane.
Show a hint
Subtract ((n·P−5)/|n|²)n.
Reveal answer and explanation
(1/3,2/3,5/3)
The correction is (2/3)n, and the resulting point satisfies the plane equation.
Practice 2
Does multiplying the plane equation by ten change the distance?
Show a hint
Numerator and denominator scale together.
Reveal answer and explanation
No
The common absolute scale factor cancels.
Take the idea with you
Translate a sensor's plane-equation error into an actual perpendicular offset.
04 · Reflect and continue
Can you explain it in your own words?
Before moving on, explain the main idea without looking at the worked example. Try both practice questions, check your reasoning, and name one mistake you now know how to avoid. Return to a step if you still need support.
Up next: Decide whether spatial lines intersect
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