Hint
Both direction vectors are nonzero. Solve the coordinate equations with separate t and u.
Worked solution
(a)
The coordinate equations give \(\displaystyle t=2, u=1\). Substitution in all three components gives the same position vector.
Answer: \(\left[\begin{matrix}15\\6\\11\end{matrix}\right]\)
(b)
Use the absolute scalar product of the two direction vectors in the cosine formula.
Answer: \(\displaystyle \operatorname{acos}{\left(\frac{13}{30} \right)}\)
(c)
Substitute either recovered parameter.
\(15\)
Answer: \(15\)
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) · 2 marks
- 1 mark: Solve using independent line parameters and verify the third coordinate.
- 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: Do not force the two line parameters to be equal.
(b) · 2 marks
- 1 mark: Use the absolute scalar product of nonzero directions divided by their magnitudes.
- 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: Matching only two coordinates does not prove a 3D intersection.
(c) · 2 marks
- 1 mark: Substitute either recovered 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: Do not force the two line parameters to be equal.
Total: 6 marks.