LEARN · EXPLAIN · REVISE

Find a nonzero vector perpendicular to both a and b, with third component 1.

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

高二選擇性必修第一册(A版).pdf · 1.3 · PDF 21 / printed page 16

TOPIC 01

Coordinate representation of space vectors

Calculate spatial lengths, angles, projections and perpendicular directions from coordinates.

Calculate spatial lengths, angles, projections and perpendicular directions from coordinates.
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

a=(1,0,4),\ b=(0,1,2)

Projection requires a nonzero direction; signed scalar projection can be negative.

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.

u\cdot a=u\cdot b=0

Worked solution

Use the dot product to connect coordinates with length or angle.
Apply the stated relation and retain its conditions.
$x+4z=0,\quad y+2z=0$
Apply the stated relation and retain its conditions.
$z=1\Rightarrow x=-4,\ y=-2$
Two linear constraints determine a perpendicular direction up to scale.

The requested relation or conclusion is shown below.

u=(-4,-2,1)

Checks and common pitfalls: Two linear constraints determine a perpendicular direction up to scale.

Reasoning checklist · self / teacher assessment

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

Teacher preparation and assessment

Calculate spatial lengths, angles, projections and perpendicular directions from coordinates.
Which condition is essential in coordinate representation of space vectors?
Predict what happens to projection when the direction vector is reversed.

Coordinates: Subtract point coordinates to obtain a displacement vector.
$\overrightarrow{AB}=B-A$
Projection conditions: Projection requires a nonzero direction; signed scalar projection can be negative.
$\operatorname{proj}_{u}v=\frac{v\cdot u}{u\cdot u}u$