LEARN · EXPLAIN · REVISE

For perpendicular vectors of lengths 3 and 4, compare |a+b| with |a|+|b|.

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 / Standard#Your turn

\mathbf a\perp\mathbf b,\quad|\mathbf a|=3,\ |\mathbf b|=4

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Use the zero cross term for perpendicular vectors.

Hint 2

Compare the norm after addition with the sum of norms.

Worked solution

Use the zero cross term for perpendicular vectors.
Calculate or simplify this relation.
$|\mathbf a+\mathbf b|^2=9+16=25;\quad|\mathbf a|+|\mathbf b|=7$
The triangle inequality may be strict.

5<7.

Checks and common pitfalls: The triangle inequality may be strict.

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.