g10-14-04 · Additional mathematics
Perpendicular to two parallel lines is not enough
Why must the two lines in the perpendicularity test intersect?
1 MIN · PREDICT ON PAPER
In the floor plane, draw two parallel x-directed lines and a y-directed line perpendicular to both. Is it perpendicular to the floor?
Discuss or write first. No typed answer is required to continue.
Independent tests force the two horizontal components to zero.
2 MIN · EXPLAIN AND REVISE
Which condition does your claim need?
Compare at least two cases. Write a reason, then check the explanation together.
Reveal explanation and conditions
A line perpendicular to two intersecting lines of a plane is perpendicular to every plane direction. Two parallel lines repeat the same test and do not control the other direction.
Conditions: Use a nonzero candidate direction and two genuinely intersecting plane lines.
Mathematical justification: For plane z=0, perpendicularity to independent x,y directions forces vx=vy=0. A nonzero remaining vz is normal to the plane. Repeating x only leaves vy unrestricted.
1 MIN · EXIT CHECK
Which is sufficient: two parallel lines or two intersecting lines in the plane?
Check after answering
Two intersecting lines, with a nonzero candidate direction.
Teacher guidance and extension
Before: Points, lines, planes and vector directions.
Ask: Does the second line test a new independent direction?
Watch for: Using two parallel floor lines as a complete perpendicularity test.
Challenge: Find a nonzero vector perpendicular to(1,0,0) that is not perpendicular to z=0.
Suggested timing: predict 1 min, compare 3 min, explain 2 min, exit check 1 min. Observations can suggest a claim; they do not prove it for every input.
Optional local notebook
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