Learn with Amar
Teaching video
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Undergraduate chapters and video availability01 · Read and understand
What you will learn
- Diagonalize a quadratic curve using principal directions.
- Justify the conclusion "u²=1, so the curve is two parallel lines" using the stated assumptions.
Before you start
Eigenvectors and quadratic forms.
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
Identify x²+2xy+y²=2 after an orthonormal coordinate change.
Why this math matters
Diagonalize a quadratic curve using principal directions. 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 coordinate transformation is orthonormal.
- The equation describes a set, not a filled inequality region.
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.
Remove a cross term by rotating coordinates
PausedQuestion: Start with the question. Paused.
Question
Start with the question
Identify x²+2xy+y²=2 after an orthonormal coordinate change.
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
u=(x+y)/√2 and v=(x−y)/√2
These coordinates follow perpendicular sum and difference directions.
Work through the mathematics
x²+2xy+y²=(x+y)²=2u²
The cross term disappears because the quadratic form has only one nonzero principal coefficient.
Check the conclusion
u²=1, so the curve is two parallel lines
A quadratic equation can be degenerate rather than an ellipse.
The result
u²=1, so the curve is two parallel lines
A quadratic equation can be degenerate rather than an ellipse.
Common mistakes to catch
- Not every quadratic curve is a nondegenerate conic.
- A rotation changes coordinates without changing Euclidean distances.
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
Rewrite x²−2xy+y²=8.
Show a hint
Use the v coordinate.
Reveal answer and explanation
v²=4
The equation becomes 2v²=8, again two parallel lines.
Practice 2
Why is this not a circle?
Show a hint
Check dependence on both rotated coordinates.
Reveal answer and explanation
v is unrestricted
A circle bounds displacement in every direction, but these lines extend indefinitely.
Take the idea with you
Use principal axes to reveal hidden degeneracy in a quadratic model.
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: Calculate length along a parameterized curve
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