LEARN · EXPLAIN · REVISE

Simplify using the triangle rule.

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

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

TOPIC 01

Operations on plane vectors

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

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

01 / Foundation#Your turn

\overrightarrow{AB}+\overrightarrow{BC}+\overrightarrow{CA}

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

Follow the closed path.

Worked solution

Use the vector operation law and preserve vector versus scalar types.
Calculate or simplify this relation.
$\overrightarrow{AB}+\overrightarrow{BC}=\overrightarrow{AC};\quad\overrightarrow{AC}+\overrightarrow{CA}=\mathbf0$
Returning to the start gives zero total displacement.

The zero vector.

Checks and common pitfalls: Returning to the start gives zero total displacement.

Reasoning checklist · self / teacher assessment

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

Teacher preparation and assessment

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.

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.