<p>Relative to the origin \(O\), the points \(A\) and \(B\) have non-zero and non-parallel position vectors \(\mathbf{a}\) and \(\mathbf{b}\) respectively. The plane \(p\) has equation \(\mathbf{r} \cdot \mathbf{n} = 0\).</p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">Given that \(\mathbf{a} \cdot \mathbf{n} = \mathbf{b} \cdot \mathbf{n} \neq 0\), show that \(\vec{AB}\) is perpendicular to \(\mathbf{n}\). Hence, describe the geometrical relationship between \(\vec{AB}\) and the plane \(p\).</span><span class="inline-marks">[2]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Write down the equation of a line parallel to \(\vec{AB}\) that is contained in the plane \(p\).</span><span class="inline-marks">[1]</span></div></li><li><div class="list-item-content"><span class="list-item-text">The point \(F\) is the foot of perpendicular from a point \(C\) with position vector \(\mathbf{c}\) to the plane \(p\). Find the position vector of point \(F\), giving your answer in terms of \(\mathbf{c}\) and \(\mathbf{n}\).</span><span class="inline-marks">[3]</span></div></li></ol>

Relative to the origin \(O\), the points \(A\) and \(B\) have non-zero and non-parallel position vectors \(\mathbf{a}\) and \(\mathbf{b}\) respectively. The plane \(p\) has equation \(\mathbf{r} \cdot \mathbf{n} = 0\).

Given that \(\mathbf{a} \cdot \mathbf{n} = \mathbf{b} \cdot \mathbf{n} \neq 0\), show that \(\vec{AB}\) is perpendicular to \(\mathbf{n}\). Hence, describe the geometrical relationship between \(\vec{AB}\) and the plane \(p\).

Write down the equation of a line parallel to \(\vec{AB}\) that is contained in the plane \(p\).

The point \(F\) is the foot of perpendicular from a point \(C\) with position vector \(\mathbf{c}\) to the plane \(p\). Find the position vector of point \(F\), giving your answer in terms of \(\mathbf{c}\) and \(\mathbf{n}\).

Vectors

International Baccalaureate Higher Level Mathematics

2025 ACJC P1 Q5

2025 ACJC P1 Q5