Acute angle between two planes

9 \includegraphics[max width=\textwidth, alt={}, center]{98ee8d3e-9aba-46a2-aa9c-b1e2093f393e-14_666_703_260_721} The diagram shows a set of rectangular axes \(O x , O y\) and \(O z\), and four points \(A , B , C\) and \(D\) with position vectors \(\overrightarrow { O A } = 3 \mathbf { i } , \overrightarrow { O B } = 3 \mathbf { i } + 4 \mathbf { j } , \overrightarrow { O C } = \mathbf { i } + 3 \mathbf { j }\) and \(\overrightarrow { O D } = 2 \mathbf { i } + 3 \mathbf { j } + 5 \mathbf { k }\).

1. Find the equation of the plane \(B C D\), giving your answer in the form \(a x + b y + c z = d\).
2. Calculate the acute angle between the planes \(B C D\) and \(O A B C\).

9. Two planes, \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\), are defined by $$\begin{aligned} & \Pi _ { 1 } : 4 x - 3 y + 2 z = 5 \\ & \Pi _ { 2 } : 6 x + y + z = 9 \end{aligned}$$

1. Find the acute angle between the planes \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\). Give your answer correct to three significant figures.
2. Find the perpendicular distance from the point \(A ( 5 , - 2 , - 6 )\) to the plane \(\Pi _ { 1 }\).
3. 1. Show that the point \(B ( 5,5,0 )\) lies on \(\Pi _ { 1 }\) and that the point \(C ( 1,3,0 )\) lies on \(\Pi _ { 2 }\).
   2. State an equation of a plane that contains the points \(B\) and \(C\).
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