LEARN · EXPLAIN · REVISE

Find λ so that the third vector lies in the plane spanned by the first two.

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 / Foundation#Your turn

a=(1,0,0),\ b=(0,1,0),\ c=(8,2,\lambda)

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.

c=xa+yb

Worked solution

Use vector components and the distributive laws.
Apply the stated relation and retain its conditions.
$c=(x,y,0)\Rightarrow\lambda=0$
The plane spanned by a and b has zero third coordinate.

The requested value is 0.

Checks and common pitfalls: The plane spanned by a and b has zero third coordinate.

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.