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Find cos θ between a and 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 / Standard#Worked example

a=(3,0,0),\quad b=(3,3,0)

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.

\cos\theta=(a\cdot b)/(|a||b|)

Worked solution

Use the dot product to connect coordinates with length or angle.
Apply the stated relation and retain its conditions.
$a\cdot b=9,\quad |a|=3,\quad |b|=3\sqrt2$
Apply the stated relation and retain its conditions.
$\cos\theta=1/\sqrt2$
A common positive scale cancels in the angle formula.

The requested value is 0.707106781187.

Checks and common pitfalls: A common positive scale cancels in the angle formula.

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.