LEARN · EXPLAIN · REVISE

Find the representation of v in the given basis.

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 / Standard#Your turn

e_1=(1,1,0),\ e_2=(1,-1,0),\ e_3=(0,0,1),\ v=(22,2,3)

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.

x+y=2t,\quad x-y=2

Worked solution

Use vector components and the distributive laws.
Apply the stated relation and retain its conditions.
$2x=24\Rightarrow x=12$
Apply the stated relation and retain its conditions.
$y=10,\quad z=3$
Solve the simultaneous component equations; uniqueness follows from independence.

The requested relation or conclusion is shown below.

v=12e_1+10e_2+3e_3

Checks and common pitfalls: Solve the simultaneous component equations; uniqueness follows from independence.

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.