A tram travels from station \(S\) to station \(T\), given by \(\vec{ST} = \begin{pmatrix} 5 \\ -2 \end{pmatrix}\), then from \(T\) to station \(U\), given b...
Question 1 Report
A tram travels from station \(S\) to station \(T\), given by \(\vec{ST} = \begin{pmatrix} 5 \\ -2 \end{pmatrix}\), then from \(T\) to station \(U\), given by \(\vec{TU} = \begin{pmatrix} -3 \\ 6 \end{pmatrix}\), with distances in kilometres.
Work out \(\vec{SU}\), the tram's direct journey from \(S\) to \(U\). (2)
Answer Details
When a journey is made in two stages, the overall direct journey (the "resultant" vector) is found simply by adding the two stage vectors component by component - the individual path taken does not matter, only the start and end points.
(a) Adding the two stages of the journey component by component: \(\vec{SU} = \vec{ST} + \vec{TU} = \begin{pmatrix} 5 \\ -2 \end{pmatrix} + \begin{pmatrix} -3 \\ 6 \end{pmatrix} = \begin{pmatrix} 5+(-3) \\ -2+6 \end{pmatrix} = \begin{pmatrix} 2 \\ 4 \end{pmatrix}\). [2 marks]
This is the triangle law for vectors: travelling from \(S\) to \(T\) and then \(T\) to \(U\) ends up in exactly the same place as travelling directly from \(S\) to \(U\), so the two stage vectors must add to give the direct route.
Download The App On Google Playstore
Everything you need to excel in your exams
