LEARN · EXPLAIN · REVISE

Find the coefficient of e1 when e1=(1,0,0), e2=(1,1,0), e3=(0,0,1).

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

TOPIC 01

Space vectors and solid geometry: mixed review

A mixed assessment: identify the method, justify it and revise your reasoning.

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

01 / Foundation#Your turn

v=(27,2,3)

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,y,z)=v

Worked solution

Use vector components and the distributive laws.
Apply the stated relation and retain its conditions.
$y=2,\quad z=3,\quad x=27-2=25$
Changing the basis changes coefficients even when the geometric vector is unchanged.

The requested value is 25.

Checks and common pitfalls: Changing the basis changes coefficients even when the geometric vector is unchanged.

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

Teacher preparation and assessment