2006 JJC P1 Q12

The position vectors of the points \(A\) and \(B\) are \(\mathbf{i}+2\mathbf{j}+3\mathbf{k}\) and \(2\mathbf{i}+3\mathbf{j}+p\mathbf{k}\) respectively, where \(p\) is a constant. The point \(C\) is such that \(OABC\) is a rectangle, where \(O\) is the origin.

1. Show that \(p=2\).
2. Write down the position vector of \(C\).
3. Find a vector equation of the line \(BC\).

The equation of line \(l\) is given by \(\frac{x-1}{3}=\frac{y-1}{2}\), \(z=1\).

4. Show that the lines \(BC\) and \(l\) are skew.