Hint
All positions use one origin. Subtract the endpoint x-coordinates.
Worked solution
(a)
Use b−a, its positive magnitude, and then normalise.
Answer: \(\left[\begin{matrix}1\\3\\7\end{matrix}\right]\); \(\sqrt{59}\); \(\displaystyle \left[\begin{matrix}\frac{\sqrt{59}}{59}\\\frac{3 \sqrt{59}}{59}\\\frac{7 \sqrt{59}}{59}\end{matrix}\right]\)
(b)
Subtract the endpoint z-coordinates.
\(7\)
Answer: \(7\)
(c)
Include all three squared components.
\(59\)
Answer: \(59\)
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: Subtract the three endpoint components in order.
- 1 mark: Use the three-dimensional magnitude and normalise the displacement.
- 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 omit the third squared component.
(b) · 2 marks
- 1 mark: Subtract the endpoint z-coordinates.
- 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: A displacement reverses sign when its endpoints are reversed.
(c) · 2 marks
- 1 mark: Include all three squared components.
- 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 omit the third squared component.
Total: 7 marks.