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Nov 2021 p33 q8
2133
In the diagram, \(OABCD\) is a pyramid with vertex \(D\). The horizontal base \(OABC\) is a square of side 4 units. The edge \(OD\) is vertical and \(OD = 4\) units. The unit vectors \(\mathbf{i}, \mathbf{j}\) and \(\mathbf{k}\) are parallel to \(OA, OC\) and \(OD\) respectively.
The midpoint of \(AB\) is \(M\) and the point \(N\) on \(CD\) is such that \(DN = 3NC\).
(a) Find a vector equation for the line through \(M\) and \(N\).
(b) Show that the length of the perpendicular from \(O\) to \(MN\) is \(\frac{1}{3}\sqrt{82}\).
Solution
(a) The position vector of \(M\) is \(\mathbf{OM} = 4\mathbf{i} + 2\mathbf{j}\).
For \(N\), since \(DN = 3NC\), \(N\) divides \(CD\) in the ratio 3:1. The position vector of \(C\) is \(4\mathbf{j}\) and \(D\) is \(4\mathbf{k}\). Using the section formula, \(\mathbf{ON} = \frac{1}{4}(3 \times 4\mathbf{k} + 1 \times 4\mathbf{j}) = 3\mathbf{j} + \mathbf{k}\).
