Points \(A , B\) and \(C\) have coordinates \(( 0,1 , - 4 ) , ( 1,1 , - 2 )\) and \(( 3,2,5 )\) respectively.
Find the vector product \(\overrightarrow { A B } \times \overrightarrow { A C }\).
Hence find the equation of the plane \(A B C\) in the form \(a x + b y + c z = d\). [0pt]
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The equation of the curve shown on the graph is, in polar coordinates, \(r = 3 \sin 2 \theta\) for \(0 \leqslant \theta \leqslant \frac { 1 } { 2 } \pi\).
\includegraphics[max width=\textwidth, alt={}, center]{25055c9c-2d29-476e-887a-a10699814b85-04_505_704_348_292}
The greatest value of \(r\) on the curve occurs at the point \(P\).
Show that \(\theta = \frac { 1 } { 4 } \pi\) at the point \(P\).
Find the value of \(r\) at the point \(P\).
Mark the point \(P\) on a copy of the graph.
In this question you must show detailed reasoning.
Find the exact area of the region enclosed by the curve. [0pt]
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