LEARN · EXPLAIN · REVISE

Expand the squared norm and explain the cross term.

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

Return to the lesson / paper ↗

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

Revisit first: Concept of plane vectors

TOPIC 01

Operations on plane vectors

Build understanding of operations on plane vectors through definitions, contrasting cases and justified applications.

What you will be able to explain

Connect geometric and algebraic forms of operations on plane vectors.
Connect representations and justify the steps, including boundary cases.
Explain a solution and apply the idea to a changed situation.

Try it. Leave your reasoning visible.

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

01 / Standard#Your turn

Expand the squared norm and explain the cross term.

|\mathbf a+\mathbf b|^2

Use the domain, units and sampling assumptions stated in the question.

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Use the vector operation law and preserve vector versus scalar types.

Hint 2

Use distributivity of the dot product.

Worked solution

Use the vector operation law and preserve vector versus scalar types.
Calculate or simplify this relation.
$(\mathbf a+\mathbf b)\cdot(\mathbf a+\mathbf b)=|\mathbf a|^2+2\mathbf a\cdot\mathbf b+|\mathbf b|^2$
The cross term vanishes exactly when the dot product is zero; this includes zero-vector cases without assigning them an angle.

|a|²+2a·b+|b|².

Checks and common pitfalls: The cross term vanishes exactly when the dot product is zero; this includes zero-vector cases without assigning them an angle.

Reasoning checklist · self / teacher assessment

State the relevant definition, condition or model.
Show a valid calculation, proof or counterexample.
Interpret the conclusion with its restrictions.

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

Teacher preparation and assessment

Question sequence

What must be true before using the main rule for operations on plane vectors?
Which representation makes this task easier, and why?
Change one assumption. Does the conclusion survive?

Board plan

Connect geometric and algebraic forms of operations on plane vectors.
Addition and scaling: Combine displacements by the triangle rule and scale length by absolute scalar value.
Dot product: The dot product is a scalar measuring directional alignment.
$\mathbf a\cdot\mathbf b=|\mathbf a||\mathbf b|\cos\theta$
Close with: conditions → representation → reasoning → check.

Anticipated thinking

Expected reasoning: Combine displacements by the triangle rule and scale length by absolute scalar value.
Expected reasoning: The dot product is a scalar measuring directional alignment.
Expected correction: Dot product and scalar multiplication have different types.

Assessment checklist

1: identify the givens and required quantity.
1: choose a valid definition, representation or method.
1: present connected, correct reasoning.
1: check conditions and explain the result.