LEARN · EXPLAIN · REVISE

Find the line through P with direction u.

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

T06高三理組數學思維本(2026).pdf · Spatial analytic geometry; also PDF10–11 cross product · PDF 15 / printed page 14

TOPIC 01

Spatial lines, planes, spheres and cross products

Construct spatial equations and use normals, cross products and direction tests.

Construct spatial equations and use normals, cross products and direction tests.
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#Your turn

P=(1,2,7),\quad u=(2,-1,3)

Directions and normals must be nonzero; parallel directions do not prove that two spatial lines coincide.

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Write the defining equation, retain exclusions, and then solve or compare.

Hint 2

Use this intermediate relation.

X=P+su

Worked solution

The requested relation or conclusion is shown below.

(x,y,z)=(1,2,7)+s(2,-1,3)

Checks and common pitfalls: One parameter controls all coordinates simultaneously.

Reasoning checklist · self / teacher assessment

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

Teacher preparation and assessment

Construct spatial equations and use normals, cross products and direction tests.
Which condition is essential in spatial lines, planes, spheres and cross products?
Can two nonparallel space lines have no intersection?

Model or definition: Construct spatial equations and use normals, cross products and direction tests.
$\ell:P+su;\quad\Pi:n\cdot(X-P)=0$
Conditions: Directions and normals must be nonzero; parallel directions do not prove that two spatial lines coincide.