Parallelism and perpendicularity in space · Teacher reference · answers included

← Senior 1 · short mathematical inquiries

Teacher reference: contains answers, conditions and proofs.

14 Parallelism and perpendicularity in space · g10-14-01 · 5–10 min · TEACHER · ANSWERS

Not intersecting does not imply parallel

Can two space lines neither intersect nor run parallel?

Predict: Compare AB with an upper parallel edge,an adjacent edge and a noncoplanar vertical edge.

Compare	Computed observation
Parallel Second line = Upper parallel edge	Rotate the view; the mathematical relation stays fixed.
Intersecting Second line = Adjacent edge	Rotate the view; the mathematical relation stays fixed.
Skew Second line = Noncoplanar edge	Rotate the view; the mathematical relation stays fixed.

Conclusion: Space lines can intersect,be parallel,or be skew. Skew lines are nonparallel and noncoplanar, even if their projections cross.

Conditions: The objects are full lines with nonzero directions, not only the visible edge segments.

Justification: Nonparallel lines are coplanar exactly when(q−p)·(d×e)=0; if the scalar triple product is nonzero, no common plane or intersection exists.

Ask: Does rotating the picture preserve the spatial relation while moving its apparent crossing?

Watch for: Inferring an actual intersection from a crossing in the flat picture.

Extension: Name a pair of cube edges that are skew.

Exit: Does “two lines do not meet” prove they are parallel in space?

No; they may be skew.

Before: Points, lines, planes and vector directions. · Timing: predict 1, compare 3, explain 2, exit 1 minute. Finite observations do not prove a universal claim.

14 Parallelism and perpendicularity in space · g10-14-02 · 5–10 min · TEACHER · ANSWERS

A parallel line must lie outside the plane

Can the outside-plane condition be omitted?

Predict: Keep a line direction parallel to the floor and lower its height to0. Predict its relation to the floor.

Compare	Computed observation
Outside parallel Starting height h = 1; Vertical direction k = 0	The direction is parallel and every point remains outside.
Contained boundary Starting height h = 0; Vertical direction k = 0	The outside-plane condition fails.
Tilted intersection Starting height h = 1; Vertical direction k = -1	A nonzero vertical direction gives one intersection.

Conclusion: A line parallel to a direction inside a plane can either lie in that plane or remain outside it. Strict line-plane parallelism requires no intersection.

Conditions: To use the criterion, the line must lie outside the plane and be parallel to a line within it.

Justification: For plane z=0 and line(t,0,h+kt), k=0,h≠0 means no meeting; k=h=0 means containment; k≠0 gives one meeting at t=−h/k.

Ask: Which parameter changes direction, and which only moves the line up or down?

Watch for: Calling a contained line parallel to its own plane.

Extension: Construct a line parallel to a plane direction but intersecting the plane.

Exit: A line with points(t,0,1) has what relation to z=0?

Parallel and outside: z remains1 for every t.

Before: Points, lines, planes and vector directions. · Timing: predict 1, compare 3, explain 2, exit 1 minute. Finite observations do not prove a universal claim.

14 Parallelism and perpendicularity in space · g10-14-03 · 5–10 min · TEACHER · ANSWERS

Two test lines must intersect

Do two parallel test lines prove that two planes are parallel?

Predict: Tilt an upper plane about one horizontal direction. Predict whether two parallel horizontal lines detect the tilt.

Compare	Computed observation
Parallel planes Plane offset h = 1; Slope in x = 0; Slope in y = 0	Two independent directions and distinct planes are required.
One-direction counterexample Plane offset h = 1; Slope in x = 0; Slope in y = 1	Both parallel x test lines pass, yet the planes intersect.
Coincident planes Plane offset h = 0; Slope in x = 0; Slope in y = 0	Coincidence is excluded from strict parallelism.

Conclusion: Two intersecting lines in one plane, each parallel to the other plane, determine two independent directions. Two parallel test lines determine only one direction and can miss a tilt.

Conditions: The planes must be distinct. Coincident planes are not strictly parallel in the no-intersection convention used here.

Justification: For z=h+sx x+sy y, both independent direction vectors(1,0,sx),(0,1,sy) are horizontal exactly when sx=sy=0. A nonzero h then separates the planes.

Ask: Which direction is never checked if both test lines run the same way?

Watch for: Replacing “two intersecting lines” with any two lines.

Extension: Use z=y+1 to produce a counterexample with two horizontal parallel lines.

Exit: Must the two lines in a plane-plane parallelism test intersect?

Yes, to supply two independent directions; also exclude coincident planes.

Before: Points, lines, planes and vector directions. · Timing: predict 1, compare 3, explain 2, exit 1 minute. Finite observations do not prove a universal claim.

14 Parallelism and perpendicularity in space · g10-14-04 · 5–10 min · TEACHER · ANSWERS

Perpendicular to two parallel lines is not enough

Why must the two lines in the perpendicularity test intersect?

Predict: In the floor plane, draw two parallel x-directed lines and a y-directed line perpendicular to both. Is it perpendicular to the floor?

Compare	Computed observation
Normal direction Direction x = 0; Direction y = 0; Direction z = 1; Test lines = Intersecting x and y lines	Independent tests force the two horizontal components to zero.
Parallel-test counterexample Direction x = 0; Direction y = 1; Direction z = 0; Test lines = Two parallel x lines	Counterexample: repeated parallel tests miss another plane direction.
Zero direction is invalid Direction x = 0; Direction y = 0; Direction z = 0; Test lines = Intersecting x and y lines	A zero vector does not define a line direction.

Conclusion: 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.

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.

Ask: Does the second line test a new independent direction?

Watch for: Using two parallel floor lines as a complete perpendicularity test.

Extension: Find a nonzero vector perpendicular to(1,0,0) that is not perpendicular to z=0.

Exit: Which is sufficient: two parallel lines or two intersecting lines in the plane?

Two intersecting lines, with a nonzero candidate direction.

Before: Points, lines, planes and vector directions. · Timing: predict 1, compare 3, explain 2, exit 1 minute. Finite observations do not prove a universal claim.

14 Parallelism and perpendicularity in space · g10-14-05 · 5–10 min · TEACHER · ANSWERS

Perpendicular planes do not make every line normal

If two planes are perpendicular, is every line of one perpendicular to the other?

Predict: Fold a page to90°. Compare the crease direction with a line across the upright page.

Compare	Computed observation
Across a right-angle fold Fold angle (degrees) = 90; Line direction within page (degrees) = 90	A plane relation does not replace checking the selected line.
Along the hinge Fold angle (degrees) = 90; Line direction within page (degrees) = 0	The planes are perpendicular, but this line is not normal to the floor.
A45° fold Fold angle (degrees) = 45; Line direction within page (degrees) = 90	A plane relation does not replace checking the selected line.

Conclusion: Perpendicular planes contain some lines normal to the other plane, but the common hinge line lies in both planes. Other directions can be oblique.

Conditions: Use the angle between the planes; at fold0° or180° the planes coincide. A candidate line must be tested separately.

Justification: Let the hinge be x and the folded direction v=(0,cosα,sinα). The planes are perpendicular at α=90°, but the hinge x remains horizontal; only suitable transverse lines are vertical.

Ask: What does the common intersection line immediately disprove?

Watch for: Changing a statement about planes into a statement about every line in a plane.

Extension: Find a line in the upright plane that makes45° with the floor.

Exit: In two perpendicular planes, is their intersection line perpendicular to either plane?

No; it lies in both planes.

Before: Points, lines, planes and vector directions. · Timing: predict 1, compare 3, explain 2, exit 1 minute. Finite observations do not prove a universal claim.