LEARN · EXPLAIN · REVISE

Operations on plane vectors

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

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.

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

PREDICT → EXPLORE → EXPLAIN → TRANSFER

Make a prediction, then explore the relationship.

Lesson question: Before calculating, predict how the conclusion changes when one defining condition in operations on plane vectors changes. Record a reason.

Model exploration: predict which displayed result changes with the parameters, check the values, and compare the observation with the lesson question.

v=(2,1); |v|=2.2361; v·(2,1)=5.

Explain: Compare two admissible cases and one boundary or invalid case. Explain the observed difference using the stated definition.

Transfer: Construct a new example and a tempting incorrect solution. Repair the solution by naming the missing condition.

Try it. Leave your reasoning visible.

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

01 / Foundation#Worked example

Find the x-coordinate of a+b.

\mathbf a=(2,2),\quad\mathbf b=(3,-1)

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

Add corresponding coordinates.

Worked solution

Use the vector operation law and preserve vector versus scalar types.
Calculate or simplify this relation.
$\mathbf a+\mathbf b=(5,1)$
Vector addition combines displacements component by component.

The requested value is 5.

Checks and common pitfalls: Vector addition combines displacements component by component.

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

02 / Standard#Worked example

Calculate the dot product.

\mathbf a=(2,2),\quad\mathbf b=(3,2)

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

Multiply corresponding coordinates and add.

Worked solution

Use the vector operation law and preserve vector versus scalar types.
Calculate or simplify this relation.
$\mathbf a\cdot\mathbf b=3(2)+2(2)=10$
The result is a scalar, not a vector.

The requested value is 10.

Checks and common pitfalls: The result is a scalar, not a vector.

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

03 / Transfer#Worked example

Is vector dot product associative as (a·b)·c=a·(b·c)? Explain the type issue.

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

Identify the type of the inner result first.

Worked solution

Use the vector operation law and preserve vector versus scalar types.
Calculate or simplify this relation.
$\mathbf a\cdot\mathbf b\in\mathbb R$
Scalar multiplication and vector dot product are different operations.

No: a·b is a scalar, so a second vector dot product is not defined in that expression.

Checks and common pitfalls: Scalar multiplication and vector dot product are different operations.

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.

04 / Foundation#Your turn

Find the x-coordinate of a+b.

\mathbf a=(3,2),\quad\mathbf b=(3,-1)

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

Add corresponding coordinates.

Worked solution

Use the vector operation law and preserve vector versus scalar types.
Calculate or simplify this relation.
$\mathbf a+\mathbf b=(6,1)$
Vector addition combines displacements component by component.

The requested value is 6.

Checks and common pitfalls: Vector addition combines displacements component by component.

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

05 / Foundation#Your turn

Calculate the dot product.

\mathbf a=(3,2),\quad\mathbf b=(3,3)

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

Multiply corresponding coordinates and add.

Worked solution

Use the vector operation law and preserve vector versus scalar types.
Calculate or simplify this relation.
$\mathbf a\cdot\mathbf b=3(3)+2(3)=15$
The result is a scalar, not a vector.

The requested value is 15.

Checks and common pitfalls: The result is a scalar, not a vector.

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

06 / Foundation#Your turn

Find the magnitude after scalar multiplication.

|\mathbf a|=3,\quad|-3\mathbf a|

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

Take the absolute value of the multiplier.

Worked solution

Use the vector operation law and preserve vector versus scalar types.
Calculate or simplify this relation.
$|-3\mathbf a|=|-3||\mathbf a|=9$
The negative sign reverses direction but does not make length negative.

The requested value is 9.

Checks and common pitfalls: The negative sign reverses direction but does not make length negative.

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

07 / Foundation#Your turn

Simplify using the triangle rule.

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

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

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

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.

08 / Standard#Your turn

Find the angle in degrees between these nonzero vectors.

\mathbf a=(3,0),\quad\mathbf b=(0,4)

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

A zero dot product gives perpendicular nonzero vectors.

Worked solution

Use the vector operation law and preserve vector versus scalar types.
Calculate or simplify this relation.
$\mathbf a\cdot\mathbf b=0\Rightarrow\cos\theta=0$
Nonzero assumptions are required to define the angle.

The requested value is 90.

Checks and common pitfalls: Nonzero assumptions are required to define the angle.

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

09 / 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.

10 / Standard#Your turn

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

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

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 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

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.

11 / Standard#Your turn

Find the x-coordinate of a+b.

\mathbf a=(4,2),\quad\mathbf b=(3,-1)

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

Add corresponding coordinates.

Worked solution

Use the vector operation law and preserve vector versus scalar types.
Calculate or simplify this relation.
$\mathbf a+\mathbf b=(7,1)$
Vector addition combines displacements component by component.

The requested value is 7.

Checks and common pitfalls: Vector addition combines displacements component by component.

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

12 / Transfer#Your turn

Calculate the dot product.

\mathbf a=(4,2),\quad\mathbf b=(3,4)

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

Multiply corresponding coordinates and add.

Worked solution

Use the vector operation law and preserve vector versus scalar types.
Calculate or simplify this relation.
$\mathbf a\cdot\mathbf b=3(4)+2(4)=20$
The result is a scalar, not a vector.

The requested value is 20.

Checks and common pitfalls: The result is a scalar, not a vector.

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

13 / Transfer#Your turn

Find the magnitude after scalar multiplication.

|\mathbf a|=4,\quad|-3\mathbf a|

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

Take the absolute value of the multiplier.

Worked solution

Use the vector operation law and preserve vector versus scalar types.
Calculate or simplify this relation.
$|-3\mathbf a|=|-3||\mathbf a|=12$
The negative sign reverses direction but does not make length negative.

The requested value is 12.

Checks and common pitfalls: The negative sign reverses direction but does not make length negative.

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

End-of-lesson check and correction

Review your latest checked answers and explanations. A draft change requires a fresh check. Written work needs your self-assessment or a teacher’s review.

Enable JavaScript for a summary of your local work.

Choose a foundation skill to revisit ↗

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.