Hint
The direction vectors are nonproportional in these examples. Check direction proportionality and all three coordinates.
Worked solution
(a)
The x,y direction determinant is −5, so the directions are not proportional. The first two equations give \(\displaystyle t=5, u=2\); the z discrepancy is 3, which prevents an intersection.
Answer: The lines are skew.
(b)
Solve the first two coordinate equations.
\(5\)
Answer: \(5\)
(c)
Retain the independent second parameter.
\(2\)
Answer: \(2\)
Maths4U indicative marking guidance
Maths4U indicative marking guidance, not an official mark scheme. Credit complete valid alternatives and consequential reasoning; an unexplained final answer does not earn a method mark.
(a) · 3 marks
- 1 mark: Check that directions are not proportional and solve a two-coordinate candidate.
- 1 mark: Check the remaining coordinate to distinguish an intersection from skew lines.
- 1 mark: Give all results requested in this part, with the correct signs, exact values, units, branch and boundary inclusion where applicable.
Alternative methods: Any complete mathematically valid alternative method with the same restrictions and equivalent exact result is acceptable.
Common error: Nonparallel does not imply intersecting in 3D.
(b) · 2 marks
- 1 mark: Solve the first two coordinate equations.
- 1 mark: Give the correct requested result, retaining signs, units and any domain restrictions.
Alternative methods: Any complete mathematically valid alternative method with the same restrictions and equivalent exact result is acceptable.
Common error: Different line parameters are allowed at the same point.
(c) · 2 marks
- 1 mark: Retain the independent second parameter.
- 1 mark: Give the correct requested result, retaining signs, units and any domain restrictions.
Alternative methods: Any complete mathematically valid alternative method with the same restrictions and equivalent exact result is acceptable.
Common error: Nonparallel does not imply intersecting in 3D.
Total: 7 marks.