The Kalu family walks between home \(H\), shop \(S\) and bank \(B\) to save fares. In km from \(O\): \(\vec{OH} = \begin{pmatrix} 0 \\ 2 \end{pmatrix}\), \(...
Question 1 Report
The Kalu family walks between home \(H\), shop \(S\) and bank \(B\) to save fares. In km from \(O\): \(\vec{OH} = \begin{pmatrix} 0 \\ 2 \end{pmatrix}\), \(\vec{OS} = \begin{pmatrix} 5 \\ 6 \end{pmatrix}\), \(\vec{OB} = \begin{pmatrix} 9 \\ 0 \end{pmatrix}\).
- Find \(\vec{HS}\) as a column vector. (1)
- Find \(\vec{HB}\) as a column vector and its magnitude, correct to 3 significant figures. (2)
- Route \(H\) to \(S\) to \(B\) is 13.6 km. Find the distance saved by the direct route \(H\) to \(B\). (2)
Answer Details
This question tests using column vectors between named points to find a displacement and its magnitude, then comparing that direct distance with a given two-stage route.
(a) The vector from \(H\) to \(S\) is found by subtracting the position vector of \(H\) from that of \(S\):
\[\vec{HS}=\vec{OS}-\vec{OH}=\begin{pmatrix}5\\6\end{pmatrix}-\begin{pmatrix}0\\2\end{pmatrix}=\begin{pmatrix}5-0\\6-2\end{pmatrix}=\begin{pmatrix}5\\4\end{pmatrix} \text{ km <b>[1]</b>}\](b) The same method gives the vector from \(H\) to \(B\):
\[\vec{HB}=\vec{OB}-\vec{OH}=\begin{pmatrix}9\\0\end{pmatrix}-\begin{pmatrix}0\\2\end{pmatrix}=\begin{pmatrix}9\\-2\end{pmatrix} \text{ km}\]Its magnitude, the straight-line distance from home to the bank, is found by Pythagoras' theorem:
\[|\vec{HB}|=\sqrt{9^2+(-2)^2}=\sqrt{81+4}=\sqrt{85}=9.22 \text{ km (3 s.f.) <b>[2] (1 for } \sqrt{85}\text{, 1 for the rounded value)</b>}\](c) The saving is the difference between the two-stage route and the direct route, using the more precise unrounded value for \(|\vec{HB}|\) to avoid compounding rounding error:
\[13.6-\sqrt{85}=13.6-9.2195\ldots=4.38 \text{ km (3 s.f.) <b>[2]</b>}\]Subtracting the two vectors, as in parts (a) and (b), gives the actual displacement between two points regardless of the path taken to get there; comparing its magnitude with a longer, indirect route is exactly how "distance saved by a shortcut" problems are solved.
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