The position vectors of the points \(A, B\) and \(C\) with respect to an origin \(O\) are the unit vectors \(\mathbf{a}, \mathbf{b}\) and \(\mathbf{c}\) respectively. It is given that \(OB\) is perpendicular to \(OC\), \(m = \cos \widehat{AOC}\), \(n = \cos \widehat{AOB}\), and \(m+n \neq 1\). The vector \(\mathbf{r}\), where \(\mathbf{r} = \mathbf{a} + \lambda \mathbf{b} + \mu \mathbf{c}\), is perpendicular to the plane \(ABC\). Show that
\[\lambda = \dfrac{(1-m)(1+m-n)}{1-m-n}\]
and find a similar expression for \(\mu\).
Note that \(\mathbf{a}\), \(\mathbf{b}\) and \(\mathbf{c}\) are unit vectors and that two of them are perpendicular.
If \(\mathbf{r}\) is perpendicular to the plane \(ABC\), then which vectors is \(\mathbf{r}\) perpendicular to?
If two vectors are perpendicular, what can we write down algebraically?
It is further given that \(\widehat{AOB}=\widehat{AOC}=120^\circ\). The point \(N\) is the foot of the perpendicular from \(O\) on the plane \(ABC\). Find the position vector of \(N\), and deduce that the position vector of the reflection of \(O\) in the plane \(ABC\) is
\[\dfrac{4}{5}\mathbf{a}+ \dfrac{3}{5}\mathbf{b} + \dfrac{3}{5}\mathbf{c}.\]
Can we use our previous results to find \(\mathbf{n}\), the position vector of \(N\)? If we know \(\mathbf{n}\), can we reflect \(O\) in \(N\) easily?
