LEARN · EXPLAIN · REVISE

Explain why these three vectors do not give unique coefficients.

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

高二選擇性必修第一册(A版).pdf · 1.2 · PDF 16 / printed page 11

TOPIC 01

The fundamental theorem of space vectors

Recognise a basis and use unique vector coordinates and affine combinations.

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

01 / Transfer#Worked example

e_1=(1,0,0), e_2=(0,1,0), e_3=(4,1,0)

A basis must be independent. Weights representing a point in a plane sum to one.

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Use vector components and the distributive laws.

Hint 2

Use this intermediate relation.

e_3=te_1+e_2

Worked solution

The requested relation or conclusion is shown below.

e_3=4e_1+e_2

Checks and common pitfalls: The dependent triple permits more than one representation.

Reasoning checklist · self / teacher assessment

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

Teacher preparation and assessment

Basis: Three noncoplanar vectors span space uniquely.
$v=xe_1+ye_2+ze_3$
Independence and affine points: A basis must be independent. Weights representing a point in a plane sum to one.

Three vectors are not automatically a basis merely because there are three of them.