<p>For any vectors \(\mathbf{m}\) and \(\mathbf{n}\), explain why \(\mathbf{m} \cdot (\mathbf{m} \times \mathbf{n}) = 0\).<span class="inline-marks">[1]</span></p>
<p>With respect to the origin \(O\), the points \(A\), \(B\) and \(C\) have position vectors \(\mathbf{a}\), \(\mathbf{b}\) and \(\mathbf{c}\) respectively. \(O\), \(A\), \(B\) and \(C\) are non-coplanar. The point \(M\) is the mid-point of \(AC\) and \(p\) denotes the plane \(OAB\).</p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">The point \(R\) is such that&nbsp;\(MR\) is perpendicular to \(p\). Show that&nbsp;\(R\) lies on a line \(\ell\) with equation \(\mathbf{r} = k(\mathbf{a}+\mathbf{c}) + \lambda(\mathbf{a} \times \mathbf{b})\), \(\lambda \in \mathbb{R}\), where&nbsp;\(k\) is a constant to be determined.</span><span class="inline-marks">[2]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Given that&nbsp;\(\mathbf{a}\) and \(\mathbf{b}\) are unit vectors perpendicular to each other, \(\mathbf{a} \cdot \mathbf{c} = -2\) and \(\mathbf{b} \cdot \mathbf{c} = 4\), find in terms of \(\mathbf{a}\) and \(\mathbf{b}\), the position vector of the point of intersection between \(\ell\) and \(p\).</span><span class="inline-marks">[3]</span></div></li></ol>

For any vectors \(\mathbf{m}\) and \(\mathbf{n}\), explain why \(\mathbf{m} \cdot (\mathbf{m} \times \mathbf{n}) = 0\).

With respect to the origin \(O\), the points \(A\), \(B\) and \(C\) have position vectors \(\mathbf{a}\), \(\mathbf{b}\) and \(\mathbf{c}\) respectively. \(O\), \(A\), \(B\) and \(C\) are non-coplanar. The point \(M\) is the mid-point of \(AC\) and \(p\) denotes the plane \(OAB\).

The point \(R\) is such that \(MR\) is perpendicular to \(p\). Show that \(R\) lies on a line \(\ell\) with equation \(\mathbf{r} = k(\mathbf{a}+\mathbf{c}) + \lambda(\mathbf{a} \times \mathbf{b})\), \(\lambda \in \mathbb{R}\), where \(k\) is a constant to be determined.

Given that \(\mathbf{a}\) and \(\mathbf{b}\) are unit vectors perpendicular to each other, \(\mathbf{a} \cdot \mathbf{c} = -2\) and \(\mathbf{b} \cdot \mathbf{c} = 4\), find in terms of \(\mathbf{a}\) and \(\mathbf{b}\), the position vector of the point of intersection between \(\ell\) and \(p\).

Vectors

International Baccalaureate Higher Level Mathematics

2025 RVHS P1 Q6

2025 RVHS P1 Q6

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