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Teaching video
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Undergraduate chapters and video availability01 · Read and understand
What you will learn
- Orthogonalize independent vectors by subtracting a projection.
- Justify the conclusion "u₁·u₂=0; ||u₁||=√2, ||u₂||=√(3/2)" using the stated assumptions.
Before you start
Dot products and projection.
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
Apply Gram–Schmidt to v₁=(1,1,0) and v₂=(1,0,1).
Why this math matters
Orthogonalize independent vectors by subtracting a projection. 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 input vectors are independent.
- Calculations here are exact, without rounding.
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.
Create perpendicular directions without changing the span
PausedQuestion: Start with the question. Paused.
Question
Start with the question
Apply Gram–Schmidt to v₁=(1,1,0) and v₂=(1,0,1).
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
u₁=v₁; (v₂·u₁)/(u₁·u₁)=1/2
Keep the first direction and measure its contribution to the second.
Work through the mathematics
u₂=v₂−u₁/2=(1/2,−1/2,1)
Removing that component preserves the span of the pair.
Check the conclusion
u₁·u₂=0; ||u₁||=√2, ||u₂||=√(3/2)
Dividing each nonzero vector by its norm yields an orthonormal basis for the same plane.
The result
u₁·u₂=0; ||u₁||=√2, ||u₂||=√(3/2)
Dividing each nonzero vector by its norm yields an orthonormal basis for the same plane.
Common mistakes to catch
- Subtract projections onto every previous orthogonal direction.
- Normalize only after verifying the vector is nonzero.
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
Orthogonalize (1,0) and (2,3).
Show a hint
Remove the horizontal component of the second.
Reveal answer and explanation
(1,0),(0,3)
Subtract 2(1,0) from (2,3).
Practice 2
What does a zero new vector reveal in exact arithmetic?
Show a hint
The projection removed everything.
Reveal answer and explanation
Dependence on earlier vectors
The candidate was already in their span.
Take the idea with you
Explain why orthonormal sensor directions simplify coefficient calculations.
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: Fit a constant when measurements disagree
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