LEARN · EXPLAIN · REVISE

Find the cosine of the acute angle between the lines.

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

高二選擇性必修第一册(A版).pdf · 1.4 · PDF 31 / printed page 26

TOPIC 01

Applications of space vectors

Use direction and normal vectors to establish angles, distances and spatial relations.

Use direction and normal vectors to establish angles, distances and spatial relations.
Justify the method and check the conditions in a new situation.

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

01 / Foundation#Worked example

u=(1,0,0), v=(1,2,0)

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.

cosθ=|u·v|/(|u||v|)

Worked solution

The requested value is 0.4472135955.

Checks and common pitfalls: The absolute dot product selects the acute line angle.

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

Teacher preparation and assessment

Use direction and normal vectors to establish angles, distances and spatial relations.
Which condition is essential in applications of space vectors?
Predict whether translating a plane changes its direction or its distance from a fixed point.

Normals: A plane normal is perpendicular to every direction in the plane.
$n\cdot(X-P)=0$
Angles and distance: Use acute line/plane angles; normals must be nonzero and distances nonnegative.
$d=\frac{|n\cdot(P-P_0)|}{|n|}$

The line-plane angle is complementary to the angle between the line direction and plane normal.