LEARN · EXPLAIN · REVISE

Calculate a·b.

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

高二選擇性必修第一册(A版).pdf · 1.1 · PDF 7 / printed page 2

TOPIC 01

Space vectors and their operations

Add, scale and multiply space vectors; distinguish scalar and vector results.

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

01 / Foundation#Your turn

a=(8,1,2),\quad b=(2,-3,1)

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Use the dot product to connect coordinates with length or angle.

Hint 2

Use this intermediate relation.

a\cdot b=a_1b_1+a_2b_2+a_3b_3

Worked solution

The requested value is 15.

Checks and common pitfalls: A dot product is a signed scalar, not a vector.

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

Teacher preparation and assessment

Operations: Addition and scaling act component by component.
$a+b=(a_1+b_1,a_2+b_2,a_3+b_3)$
Angle conditions: The angle formula requires two nonzero vectors; a dot product is a scalar.
$\cos\theta=\frac{a\cdot b}{|a||b|}$

A zero dot product indicates perpendicular directions only when both vectors are nonzero.