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<p>Relative to the origin \(O\), the points \(A\), \(B\), and \(C\) have position vectors \(\mathbf{a}\), \(\mathbf{b}\) and \(3\mathbf{a}+\mathbf{b}\) respectively. The point \(D\) lies on \(AB\) such that \(AD=kAB\), where \(0&lt;k&lt;1\).</p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">Find a vector equation of the line \(OD\) in terms of \(k\), \(\mathbf{a}\) and \(\mathbf{b}\).</span><span class="inline-marks">[2]</span></div></li><li><div class="list-item-content"><span class="list-item-text">The point \(E\) is the midpoint of \(BC\). Find the value of \(k\) if \(O\), \(D\) and \(E\) are collinear.</span><span class="inline-marks">[4]</span></div></li></ol>
<p>It is given that \(|\mathbf{a}|=1\), \(|\mathbf{b}|=2\) and \(|3\mathbf{a}-2\mathbf{b}|=\sqrt{31}\).</p>
<ol class="abc-list" start="3"><li><div class="list-item-content"><span class="list-item-text">By considering the scalar product \((3\mathbf{a}-2\mathbf{b}) \cdot (3\mathbf{a}-2\mathbf{b})\), find the numerical value of \(\mathbf{a} \cdot \mathbf{b}\) and hence determine the angle between \(\mathbf{a}\) and \(\mathbf{b}\).</span><span class="inline-marks">[4]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Give a geometrical interpretation of \(|\mathbf{a} \cdot \mathbf{b}|\).</span><span class="inline-marks">[1]</span></div></li></ol>

Relative to the origin \(O\), the points \(A\), \(B\), and \(C\) have position vectors \(\mathbf{a}\), \(\mathbf{b}\) and \(3\mathbf{a}+\mathbf{b}\) respectively. The point \(D\) lies on \(AB\) such that \(AD=kAB\), where \(0<k<1\).

Find a vector equation of the line \(OD\) in terms of \(k\), \(\mathbf{a}\) and \(\mathbf{b}\).

The point \(E\) is the midpoint of \(BC\). Find the value of \(k\) if \(O\), \(D\) and \(E\) are collinear.

It is given that \(|\mathbf{a}|=1\), \(|\mathbf{b}|=2\) and \(|3\mathbf{a}-2\mathbf{b}|=\sqrt{31}\).

By considering the scalar product \((3\mathbf{a}-2\mathbf{b}) \cdot (3\mathbf{a}-2\mathbf{b})\), find the numerical value of \(\mathbf{a} \cdot \mathbf{b}\) and hence determine the angle between \(\mathbf{a}\) and \(\mathbf{b}\).

2025 DHS P2 Q4

2025 DHS P2 Q4