LEARN · EXPLAIN · REVISE

Explain why parallel vectors cannot form a basis for the plane.

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

高一必修第二册(A版).pdf · 6.3 · PDF 32 / printed page 25

TOPIC 01

Basis theorem and coordinate representation

Build understanding of basis theorem and coordinate representation through definitions, contrasting cases and justified applications.

Connect geometric and algebraic forms of basis theorem and coordinate representation.
Connect representations and justify the steps, including boundary cases.
Explain a solution and apply the idea to a changed situation.

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

01 / Standard#Your turn

\mathbf e_2=2\mathbf e_1

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Use a nonparallel basis or Cartesian coordinates and solve component equations.

Hint 2

Substitute the dependence relation.

Worked solution

Use a nonparallel basis or Cartesian coordinates and solve component equations.
Calculate or simplify this relation.
$a\mathbf e_1+b\mathbf e_2=(a+2b)\mathbf e_1$
They cannot represent a vector outside that line.

All their linear combinations stay on one line.

Checks and common pitfalls: They cannot represent a vector outside that line.

Reasoning checklist · self / teacher assessment

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

Teacher preparation and assessment

What must be true before using the main rule for basis theorem and coordinate representation?
Which representation makes this task easier, and why?
Change one assumption. Does the conclusion survive?

Connect geometric and algebraic forms of basis theorem and coordinate representation.
Basis: Two nonparallel vectors represent every plane vector uniquely.
Coordinate tests: Use scalar multiples for parallelism and zero dot product for perpendicular nonzero vectors.
Close with: conditions → representation → reasoning → check.

Expected reasoning: Two nonparallel vectors represent every plane vector uniquely.
Expected reasoning: Use scalar multiples for parallelism and zero dot product for perpendicular nonzero vectors.
Expected correction: Parallel basis candidates do not span the whole plane.