Hint
A plane normal and a line direction must be nonzero. Substitute the line coordinates into the plane.
Worked solution
(a)
\(\displaystyle t= \frac{c- n\cdot P}{n\cdot d}= - \frac{31}{5}\)
Answer: \(\displaystyle - \frac{31}{5}\).
Answer: \(\displaystyle - \frac{31}{5}\)
(b)
\(\displaystyle r_{\rm intersection}=\left[\begin{matrix}\frac{9}{5}\\\frac{34}{5}\\- \frac{16}{5}\end{matrix}\right]\)
Answer: \(\displaystyle \frac{9}{5}\).
Answer: \(\displaystyle \frac{9}{5}\)
(c)
\(\displaystyle \frac{|n\cdot P- c|}{|n|}= \frac{31}{3}\)
Answer: \(\displaystyle \frac{31}{3}\).
Answer: \(\displaystyle \frac{31}{3}\)
Maths4U indicative marking guidance
Maths4U indicative marking guidance, not an official mark scheme. Credit valid alternative reasoning and consequential working; do not award a reasoning mark for an unexplained final answer.
(a) · 2 marks
- 1 mark: Substitute the line coordinates into the plane.
- 1 mark: Obtain the stated result with the required sign, units and domain.
Alternative methods: Any complete mathematically valid method with the stated assumptions and equivalent result is acceptable.
Common error: Do not divide by a zero n·d.
(b) · 2 marks
- 1 mark: Substitute the parameter back into the line.
- 1 mark: Obtain the stated result with the required sign, units and domain.
Alternative methods: Any complete mathematically valid method with the stated assumptions and equivalent result is acceptable.
Common error: The distance formula uses the norm of the normal, not its squared norm.
(c) · 2 marks
- 1 mark: Divide the absolute residual by the normal length 3.
- 1 mark: Obtain the stated result with the required sign, units and domain.
Alternative methods: Any complete mathematically valid method with the stated assumptions and equivalent result is acceptable.
Common error: Do not divide by a zero n·d.
Total: 6 marks.