LEARN · EXPLAIN · REVISE

Find the plane through P with normal n.

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 / Transfer#Worked example

P=(1,2,4),\quad n=(2,-1,3)

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Translate the geometric condition into a coordinate equation.

Hint 2

Use this intermediate relation.

n\cdot(X-P)=0

Worked solution

The requested relation or conclusion is shown below.

2x-y+3z=12

Checks and common pitfalls: The constant depends on the point; the coefficients specify the normal.

Reasoning checklist · self / teacher assessment

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.