Angle between two planes

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\includegraphics[max width=\textwidth, alt={}, center]{8580dddb-cc72-4745-9e0f-1ac641c6506d-3_693_537_1206_804} The diagram shows a set of rectangular axes \(O x , O y\) and \(O z\), and three points \(A , B\) and \(C\) with position vectors \(\overrightarrow { O A } = \left( \begin{array} { l } 2 \\ 0 \\ 0 \end{array} \right) , \overrightarrow { O B } = \left( \begin{array} { l } 1 \\ 2 \\ 0 \end{array} \right)\) and \(\overrightarrow { O C } = \left( \begin{array} { l } 1 \\ 1 \\ 2 \end{array} \right)\).

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

5. The plane \(\Pi _ { 1 }\) has equation \(x - 2 y - 3 z = 5\) and the plane \(\Pi _ { 2 }\) has equation \(6 x + y - 4 z = 7\)

1. Find, to the nearest degree, the acute angle between \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\) The point \(P\) has coordinates \(( 2,3 , - 1 )\). The line \(l\) is perpendicular to \(\Pi _ { 1 }\) and passes through the point \(P\). The line \(l\) intersects \(\Pi _ { 2 }\) at the point \(Q\).
2. Find the coordinates of \(Q\). The plane \(\Pi _ { 3 }\) passes through the point \(Q\) and is perpendicular to \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\)
3. Find an equation of the plane \(\Pi _ { 3 }\) in the form \(\mathbf { r } . \mathbf { n } = p\)

6 A tetrahedron \(A B C D\) is such that \(A B\) is perpendicular to the base \(B C D\). The coordinates of the points \(A , C\) and \(D\) are \(( - 1 , - 7,2 ) , ( 5,0,3 )\) and \(( - 1,3,3 )\) respectively, and the equation of the plane \(B C D\) is \(x + 2 y - 2 z = - 1\).

1. Find, in either order, the coordinates of \(B\) and the length of \(A B\).
2. Find the acute angle between the planes \(A C D\) and \(B C D\).
3. (a) Verify, without using a calculator, that \(\theta = \frac { 1 } { 8 } \pi\) is a solution of the equation \(\sin 6 \theta = \sin 2 \theta\).
   (b) By sketching the graphs of \(y = \sin 6 \theta\) and \(y = \sin 2 \theta\) for \(0 \leqslant \theta \leqslant \frac { 1 } { 2 } \pi\), or otherwise, find the other solution of the equation \(\sin 6 \theta = \sin 2 \theta\) in the interval \(0 < \theta < \frac { 1 } { 2 } \pi\).
4. Use de Moivre's theorem to prove that $$\sin 6 \theta \equiv \sin 2 \theta \left( 16 \cos ^ { 4 } \theta - 16 \cos ^ { 2 } \theta + 3 \right) .$$
5. Hence show that one of the solutions obtained in part (i) satisfies \(\cos ^ { 2 } \theta = \frac { 1 } { 4 } ( 2 - \sqrt { 2 } )\), and justify which solution it is.

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\includegraphics[max width=\textwidth, alt={}, center]{bc975428-c594-427b-a32e-268412b3cd26-3_554_825_264_660} The cuboid \(O A B C D E F G\) shown in the diagram has \(\overrightarrow { O A } = 4 \mathbf { i } , \overrightarrow { O C } = 2 \mathbf { j } , \overrightarrow { O D } = 3 \mathbf { k }\), and \(M\) is the mid-point of \(G F\).

1. Find the equation of the plane \(A C G E\), giving your answer in the form r.n \(= p\).
2. The plane \(O E F C\) has equation \(\mathbf { r } \cdot ( 3 \mathbf { i } - 4 \mathbf { k } ) = 0\). Find the acute angle between the planes \(O E F C\) and \(A C G E\).
3. The line \(A M\) meets the plane \(O E F C\) at the point \(W\). Find the ratio \(A W : W M\).

5 A regular tetrahedron has vertices at the points $$A \left( 0,0 , \frac { 2 } { 3 } \sqrt { 6 } \right) , \quad B \left( \frac { 2 } { 3 } \sqrt { 3 } , 0,0 \right) , \quad C \left( - \frac { 1 } { 3 } \sqrt { 3 } , 1,0 \right) , \quad D \left( - \frac { 1 } { 3 } \sqrt { 3 } , - 1,0 \right) .$$

1. Obtain the equation of the face \(A B C\) in the form $$x + \sqrt { 3 } y + \left( \frac { 1 } { 2 } \sqrt { 2 } \right) z = \frac { 2 } { 3 } \sqrt { 3 }$$ (Answers which only verify the given equation will not receive full credit.)
2. Give a geometrical reason why the equation of the face \(A B D\) can be expressed as $$x - \sqrt { 3 } y + \left( \frac { 1 } { 2 } \sqrt { 2 } \right) z = \frac { 2 } { 3 } \sqrt { 3 }$$
3. Hence find the cosine of the angle between two faces of the tetrahedron.

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1. Write down normal vectors to the planes \(2 x - y + z = 2\) and \(x - z = 1\).
   Hence find the acute angle between the planes.
2. Write down a vector equation of the line through \(( 2,0,1 )\) perpendicular to the plane \(2 x - y + z = 2\). Find the point of intersection of this line with the plane.

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 \(\Pi _ { 2 }\).

10 The plane P has normal vector \(2 \mathbf { i } + a \mathbf { j } - \mathbf { k }\), where \(a\) is a positive constant, and the point ( \(3 , - 1,1\) ) lies in P . The plane \(\mathrm { x } - \mathrm { z } = 3\) makes an angle of \(45 ^ { \circ }\) with P . Find the cartesian equation of P . \section*{END OF QUESTION PAPER}

3 Three planes have the following equations. $$\begin{aligned} 2 x - 3 y + z & = - 3 \\ x - 4 y + 2 z & = 1 \\ - 3 x - 2 y + 3 z & = 14 \end{aligned}$$

1. 1. Write the system of equations in matrix form.
   2. Hence find the point of intersection of the planes.
2. In this question you must show detailed reasoning. Find the acute angle between the planes \(2 x - 3 y + z = - 3\) and \(x - 4 y + 2 z = 1\).

5 Two planes, \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\), have equations \(3 x + 2 y + z = 4\) and \(2 x + y + z = 3\) respectively.

1. Find the acute angle between \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\). The line \(L\) has equation \(x = 1 - y = 2 - z\).
2. Show that \(L\) lies in both planes.

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1. Find the acute angle between the lines \(L _ { 1 }\) and \(L _ { 2 }\), giving your answer to the nearest \(0.1 ^ { \circ }\)
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2. The lines \(L _ { 1 }\) and \(L _ { 2 }\) lie in the plane \(\Pi _ { 1 }\)
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3. 1. Find the equation of \(\Pi _ { 1 }\), giving your answer in the form r.n \(= d\)
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4. (ii) Hence find the shortest distance of the plane \(\Pi _ { 1 }\) from the origin. 11
5. The points \(A ( 4 , - 1 , - 1 ) , B ( 1,5 , - 7 )\) and \(C ( 3,4 , - 8 )\) lie in the plane \(\Pi _ { 2 }\)
   Find the angle between the planes \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\), giving your answer to the nearest \(0.1 ^ { \circ }\)