2024 CJC Promo Q12

The diagram below shows a triangular base pyramid with vertices, \(A\), \(B\), \(C\) and \(D\). With reference to the origin \(O\), the points \(A\), \(B\), \(C\) and \(D\) are \(\mathbf{i}+2\mathbf{j}+3\mathbf{k}\); \(2\mathbf{i}-\mathbf{j}+4\mathbf{k}\); \(4\mathbf{i}+5\mathbf{j}-\mathbf{k}\) and \(-2\mathbf{i}-5\mathbf{k}\) respectively.

Let the plane containing points \(A\), \(B\) and \(C\) be represented by \(\Pi \).

1. Show that the equation of \(\Pi \) can be expressed as \(\mathbf{r}\cdot \left( \begin{matrix} \begin{matrix} 9 \\ 7 \\ 12 \\ \end{matrix} \\ \end{matrix} \right)=\alpha \), where \(\alpha \) is a constant to be determined.[3]
2. Find the acute angle between line \(BD\) and \(\Pi \).[2]
3. Find the position vector of the foot of perpendicular from the point \(D\) to \(\Pi \).[4]
4. Find the area of triangle \(ABC\). Hence, find the volume of the pyramid.[4]

\(\left[ \mathrm{Volume of pyramid}=\frac{1}{3}\times \mathrm{base area}\times \mathrm{height} \right]\)