LEARN · EXPLAIN · REVISE

P=αA+βB+γC is affine, α=1/(t+1), β=1/2. Find γ.

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#Worked example

t=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

Translate the geometric condition into a coordinate equation.

Hint 2

Use this intermediate relation.

α+β+γ=1

Worked solution

The requested value is 0.25.

Checks and common pitfalls: Affine weights sum to one.

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.