4. The points \(A\) and \(B\) have position vectors \(5 \mathbf { j } + 11 \mathbf { k }\) and \(c \mathbf { i } + d \mathbf { j } + 21 \mathbf { k }\) respectively, where \(c\) and \(d\) are constants.
The line \(l\), through the points \(A\) and \(B\), has vector equation \(\mathbf { r } = 5 \mathbf { j } + 11 \mathbf { k } + \lambda ( 2 \mathbf { i } + \mathbf { j } + 5 \mathbf { k } )\), where \(\lambda\) is a parameter.
- Find the value of \(c\) and the value of \(d\).
(3)
The point \(P\) lies on the line \(l\), and \(\overrightarrow { O P }\) is perpendicular to \(l\), where \(O\) is the origin. - Find the position vector of \(P\).
(6) - Find the area of triangle \(O A B\), giving your answer to 3 significant figures.
(4)
(Total 13 marks)
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