18 In this question \(\mathbf { i }\) and \(\mathbf { j }\) are perpendicular unit vectors representing due east and due north respectively.
A particle, \(T\), is moving on a plane at a constant speed.
The path followed by \(T\) makes the exact shape of a triangle \(A B C\).
\(T\) moves around \(A B C\) in an anticlockwise direction as shown in the diagram below.
\includegraphics[max width=\textwidth, alt={}, center]{de8a7d38-a665-4feb-854e-ac83f413d133-28_447_366_671_925}
On its journey from \(A\) to \(B\) the velocity vector of \(T\) is \(( 3 \mathbf { i } + \sqrt { 3 } \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\)
18
- Find the speed of \(T\) as it moves from \(A\) to \(B\)
18
- On its journey from \(B\) to \(C\) the velocity vector of \(T\) is \(( - 3 \mathbf { i } + \sqrt { 3 } \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\) Show that the acute angle \(A B C = 60 ^ { \circ }\)
18 - It is given that \(A B C\) is an equilateral triangle.
\(T\) returns to its initial position after 9 seconds.
Vertex \(B\) lies at position vector \(\left[ \begin{array} { l } 1
0 \end{array} \right]\) metres with respect to a fixed origin \(O\)
Find the position vector of \(C\)