Hint
The stated ratio is AM:MB, in that order. Use \(\displaystyle a+ \frac{m(b- a)}{m+ n}\).
Worked solution
(a)
\(\displaystyle AM= \frac{1}{1+ 3}AB\). This multiplier is strictly between zero and one, so M lies inside the segment.
Answer: \(\displaystyle \left[\begin{matrix}\frac{31}{4}\\\frac{9}{2}\end{matrix}\right]\)
(b)
Add OA to \(\displaystyle AM= \frac{m(b- a)}{m+ n}\), then collect the a and b terms.
Answer: \(\displaystyle \overrightarrow{OM}=\frac{n\mathbf a+m\mathbf b}{m+n}\)
(c)
Collect the a terms.
\(\displaystyle \frac{3}{4}\)
Answer: \(\displaystyle \frac{3}{4}\)
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: Apply the internal division ratio to AB and add OA.
- 1 mark: Show that the displacement multiplier is strictly between zero and one.
- 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 endpoint weights are opposite the adjacent segment names.
(b) · 2 marks
- 1 mark: Write \(\displaystyle OM=OA+AM\) and collect the two endpoint-vector coefficients.
- 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: Parallel lines need not be the same line.
(c) · 2 marks
- 1 mark: Collect the a terms.
- 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: The endpoint weights are opposite the adjacent segment names.
Total: 7 marks.