Learn with Amar
Teaching video
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Undergraduate chapters and video availability01 · Read and understand
What you will learn
- Compute a least-distance projection onto a line.
- Justify the conclusion "a·r=2−2=0; ||r||²=5" 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
Project b=(3,1) onto the line spanned by a=(1,2).
Why this math matters
Compute a least-distance projection onto a line. 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 spanning vector is nonzero.
- Distance uses the ordinary Euclidean inner product.
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.
Separate a signal from its perpendicular residual
PausedQuestion: Start with the question. Paused.
Question
Start with the question
Project b=(3,1) onto the line spanned by a=(1,2).
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
c=(a·b)/(a·a)=5/5=1
The projection coefficient forces the residual to be perpendicular to the line.
Work through the mathematics
p=ca=(1,2); r=b−p=(2,−1)
Decompose the input into an allowed component and an unexplained component.
Check the conclusion
a·r=2−2=0; ||r||²=5
Orthogonality certifies that p is the closest vector on the line in Euclidean distance.
The result
a·r=2−2=0; ||r||²=5
Orthogonality certifies that p is the closest vector on the line in Euclidean distance.
Common mistakes to catch
- Do not divide by ||a|| instead of ||a||².
- Projection onto a line is different from projection onto a line segment.
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
Project (2,0) onto span(1,1).
Show a hint
Divide the dot product by two.
Reveal answer and explanation
(1,1)
The coefficient is (2+0)/(1+1)=1.
Practice 2
What is the projection of a perpendicular vector?
Show a hint
Its dot product with a is zero.
Reveal answer and explanation
The zero vector
A zero coefficient leaves the entire input in the residual.
Take the idea with you
Interpret a residual as the part of a measurement a one-direction model cannot explain.
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: Create perpendicular directions without changing the span
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