6 The points \(A , B\) and \(C\) have position vectors \(2 \mathbf { i } , 3 \mathbf { j }\) and \(4 \mathbf { k }\) respectively. Find a vector which is perpendicular to the plane \(\Pi _ { 1 }\) containing \(A , B\) and \(C\). The plane \(\Pi _ { 2 }\) has equation $$\mathbf { r } = \mathbf { i } + 4 \mathbf { j } + 2 \mathbf { k } + \lambda ( \mathbf { i } - \mathbf { j } ) + \mu ( \mathbf { j } - \mathbf { k } ) .$$ Find the acute angle between the planes \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\).

6 The points $A , B$ and $C$ have position vectors $2 \mathbf { i } , 3 \mathbf { j }$ and $4 \mathbf { k }$ respectively. Find a vector which is perpendicular to the plane $\Pi _ { 1 }$ containing $A , B$ and $C$.

The plane $\Pi _ { 2 }$ has equation

$$\mathbf { r } = \mathbf { i } + 4 \mathbf { j } + 2 \mathbf { k } + \lambda ( \mathbf { i } - \mathbf { j } ) + \mu ( \mathbf { j } - \mathbf { k } ) .$$

Find the acute angle between the planes $\Pi _ { 1 }$ and $\Pi _ { 2 }$.

\hfill \mbox{\textit{CAIE FP1 2007 Q6 [8]}}