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Find the point of segment AB closest to P; explain why unrestricted projection is insufficient.

Read the idea, work independently, then explain what changed.

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高一必修第二册(A版).pdf · 6.2 · PDF 14 / printed page 7

Revisit first: Coordinates, bases and division

TOPIC 01

Dot products and constrained projection

Connect signed projection, angle and squared distance; distinguish a line projection from a nearest point on a segment.

What you will be able to explain

Interpret the sign of a dot product.
Expand squared vector norms correctly.
Check zero vectors and segment endpoints before interpreting an angle or minimizer.

Try it. Leave your reasoning visible.

Use one hint at a time. A correction explains what changed, not just the final answer.

Examples 1–3: study a full demonstration; finish a solution from its first step; then choose a method independently before opening the solution. The remaining exercises may continue after class.

01 / Transfer#Your turn

Find the point of segment AB closest to P; explain why unrestricted projection is insufficient.

A=(0,0),B=(2,0),P=(3,4)

Skills and prerequisite lessons

Scope: JM02

Working and explanation

BUILD THE REASONING

Hint 1

First project onto the whole line.

Hint 2

Then impose the segment parameter interval.

Worked solution

The line projection has x=3, outside [0,2].
$|(x,0)-P|^2=(x-3)^2+16$
On [0,2] the square decreases, so the nearest point is the endpoint.
$Q=B=(2,0),\ PQ=\sqrt{17}$

B is nearest; distance √17.

Checks and common pitfalls: A foot outside the segment is not a feasible minimizer.

Reasoning checklist · self / teacher assessment

B is nearest; distance √17.
A foot outside the segment is not a feasible minimizer.

Think first. Reveal a hint when the class is ready.

Teacher preparation and assessment

40-minute lesson

Minutes	Focus	Teaching and student action
5	Prediction and diagnosis	Compare positive, zero and negative work.
8	Concept derivation	Derive projection by making the residual dot product zero.
10	Guided application	Compute and distinguish projection coefficient and length.
12	Transfer and explanation	Solve the nearest-segment problem and the endpoint-inclusive circle locus.
5	Exit evidence	Explain the zero-vector angle exception.

Question sequence

Why can work be negative even though distance is nonnegative?
Does a perpendicular foot always lie on the segment?

Board development

Keep coefficient t, signed projected length t|u| and vector tu in separate columns.
Sketch the external foot and highlight the admissible segment interval.

Expected correct and incorrect responses

Students divide by |u| instead of |u|² when finding t. Ask for the vector tu and check its residual dot product.
Correct: “The coefficient is 1.5, but the signed projection length is 3.”
Incorrect: “The projection coefficient and projection length are always the same.”

Follow-up questions

How does changing the length of u without changing its direction alter t and tu?

Remediation

Project (3,4) onto (2,0): label coefficient 1.5, projected length 3 and vector (3,0).

Exit check

Name the two assumptions needed before interpreting a dot-product quotient as a cosine.

Assessment checklist and sample responses

Credit expansion, admissible parameter range and geometric interpretation independently. Sample: t=3/2 is outside [0,1], so the nearest segment point occurs at t=1.
Correct: “The coefficient is 1.5, but the signed projection length is 3.”