7．The points \(O , P\) and \(Q\) lie on a circle \(C\) with diameter \(O Q\) ．The position vectors of \(P\) and \(Q\) ， relative to \(O\) ，are \(\mathbf { p }\) and \(\mathbf { q }\) respectively．
（a）Prove that \(\mathbf { p } . \mathbf { q } = | \mathbf { p } | ^ { 2 }\) ．
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\caption{Figure 3}
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The point \(R\) also lies on \(C\) and \(O P Q R\) is a kite \(K\) as shown in Figure 3．The point \(S\) has position vector，relative to \(O\) ，of \(\lambda \mathbf { q }\) ，where \(\lambda\) is a constant．Given that \(\mathbf { p } = \mathbf { i } + 2 \mathbf { j } - \mathbf { k } , \mathbf { q } = 2 \mathbf { i } + \mathbf { j } - 2 \mathbf { k }\) and that \(O Q\) is perpendicular to \(P S\) ，find
（b）the value of \(\lambda\) ，
（c）the position vector of \(R\) ，
（d）the area of \(K\) ．
Another circle \(C _ { 1 }\) is drawn inside \(K\) so that the 4 sides of the kite are each tangents to \(C _ { 1 }\) ．
（e）Find the radius of \(C _ { 1 }\) giving your answer in the form \(( \sqrt { } 2 - 1 ) \sqrt { } n\) ，where \(n\) is an integer．
A second kite \(K _ { 1 }\) is similar to \(K\) and is drawn inside \(C _ { 1 }\) ．
（f）Find that area of \(K _ { 1 }\) ．