<p>In the diagram, \(OA\) is parallel to \(CB\) and \(SC\) is parallel to \(AB\). The position vectors of \(A\) and \(B\) relative to \(O\) are \(4\mathbf{a}\) and \(5\mathbf{b}\) respectively. It is given that \(OS : SB = 2 : 3\) and \(\vec{AT} = \frac{1}{3}\vec{TB}\).</p>
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<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">Find, in terms of \(\mathbf{a}\) and \(\mathbf{b}\), the vectors</span></div><ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">\(\vec{OS}\),</span></div></li><li><div class="list-item-content"><span class="list-item-text">\(\vec{AS}\),</span></div></li><li><div class="list-item-content"><span class="list-item-text">\(\vec{OT}\).</span></div></li></ol></li><li><div class="list-item-content"><span class="list-item-text">\(U\) is a point on \(OT\) such that \(OU : OT = 8 : 3\).</span></div><ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">Express \(\vec{AU}\) in terms of \(\mathbf{a}\) and \(\mathbf{b}\).</span></div></li><li><div class="list-item-content"><span class="list-item-text">Hence write down two facts about \(A\), \(U\) and \(S\).</span></div></li></ol></li><li><div class="list-item-content"><span class="list-item-text">Prove that \(\Delta SBC\) is similar to \(\Delta BOA\).</span></div></li><li><div class="list-item-content"><span class="list-item-text">Find</span></div><ol class="i-list"><li><div class="list-item-paragraph">\(\frac{\text{Area of } \Delta SBC}{\text{Area of } \Delta BOA}\),</div></li><li><div class="list-item-paragraph">\(\frac{\text{Area of } \Delta OTA}{\text{Area of } \Delta OBA}\).</div></li></ol></li></ol>

In the diagram, \(OA\) is parallel to \(CB\) and \(SC\) is parallel to \(AB\). The position vectors of \(A\) and \(B\) relative to \(O\) are \(4\mathbf{a}\) and \(5\mathbf{b}\) respectively. It is given that \(OS : SB = 2 : 3\) and \(\vec{AT} = \frac{1}{3}\vec{TB}\).

Find, in terms of \(\mathbf{a}\) and \(\mathbf{b}\), the vectors

\(U\) is a point on \(OT\) such that \(OU : OT = 8 : 3\).

Express \(\vec{AU}\) in terms of \(\mathbf{a}\) and \(\mathbf{b}\).

Hence write down two facts about \(A\), \(U\) and \(S\).

Prove that \(\Delta SBC\) is similar to \(\Delta BOA\).

\[\frac{\text{Area of } \Delta SBC}{\text{Area of } \Delta BOA}\]

\[\frac{\text{Area of } \Delta OTA}{\text{Area of } \Delta OBA}\]

Vectors: Properties of Parallelism and Intersections

Vectors: Properties of Parallelism and Intersections

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