Hint
The line direction d is nonzero. Set (p−a−td)·d=0.
Worked solution
(a)
\(0 = 0\) verifies that PH is perpendicular to the nonzero line direction.
Answer: \(\displaystyle \left[\begin{matrix}\frac{365}{22}\\\frac{63}{11}\\\frac{167}{22}\end{matrix}\right]\)
(b)
The point-line distance is the magnitude of the perpendicular residual p−H.
Answer: \(\displaystyle \frac{\sqrt{51106}}{22}\)
(c)
Use the same parameter in every component.
\(\displaystyle \frac{63}{11}\)
Answer: \(\displaystyle \frac{63}{11}\)
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: Solve (p−a−td)·d=0 and substitute the parameter in the line.
- 1 mark: Verify that the foot is on the line and its residual is perpendicular.
- 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 substitute a coordinate of P directly as the line parameter.
(b) · 2 marks
- 1 mark: Find the magnitude of the perpendicular residual from P to H.
- 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: The residual vector must be perpendicular to d.
(c) · 2 marks
- 1 mark: Use the same parameter in every component.
- 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 substitute a coordinate of P directly as the line parameter.
Total: 7 marks.