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 \(C\).
      Additional page, if required. Write the question number(s) in the left-hand margin. Additional page, if required. Write the question number(s) in the left-hand margin. \section*{PLEASE DO NOT WRITE ON THIS PAGE}

8 The plane \(\Pi _ { 1 }\) has equation $$\mathbf { r } = \left( \begin{array} { l } 5 \\ 1 \\ 0 \end{array} \right) + s \left( \begin{array} { r } - 4 \\ 1 \\ 3 \end{array} \right) + t \left( \begin{array} { l } 0 \\ 1 \\ 2 \end{array} \right)$$

1. Find a cartesian equation of \(\Pi _ { 1 }\).
   The plane \(\Pi _ { 2 }\) has equation \(3 x + y - z = 3\).
2. Find the acute angle between \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\), giving your answer in degrees.
3. Find an equation of the line of intersection of \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\), giving your answer in the form \(\mathbf { r } = \mathbf { a } + \lambda \mathbf { b }\). [5]

The planes \(m\) and \(n\) have equations \(3x + y - 2z = 10\) and \(x - 2y + 2z = 5\) respectively. The line \(l\) has equation \(\mathbf{r} = 4\mathbf{i} + 2\mathbf{j} + \mathbf{k} + \lambda(\mathbf{i} + \mathbf{j} + 2\mathbf{k})\).

1. Show that \(l\) is parallel to \(m\). [3]
2. Calculate the acute angle between the planes \(m\) and \(n\). [3]
3. A point \(P\) lies on the line \(l\). The perpendicular distance of \(P\) from the plane \(n\) is equal to 2. Find the position vectors of the two possible positions of \(P\). [4]

A tetrahedron \(ABCD\) is such that \(AB\) is perpendicular to the base \(BCD\). 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 \(BCD\) is \(x + 2y - 2z = -1\).

1. Find, in either order, the coordinates of \(B\) and the length of \(AB\). [5]
2. Find the acute angle between the planes \(ACD\) and \(BCD\). [6]

\includegraphics{figure_6} The cuboid \(OABCDEFG\) shown in the diagram has \(\overrightarrow{OA} = 4\mathbf{i}, \overrightarrow{OC} = 2\mathbf{j}, \overrightarrow{OD} = 3\mathbf{k}\), and \(M\) is the mid-point of \(GF\).

1. Find the equation of the plane \(ACGE\), giving your answer in the form \(\mathbf{r} \cdot \mathbf{n} = p\). [4]
2. The plane \(OEFC\) has equation \(\mathbf{r} \cdot (3\mathbf{i} - 4\mathbf{k}) = 0\). Find the acute angle between the planes \(OEFC\) and \(ACGE\). [4]
3. The line \(AM\) meets the plane \(OEFC\) at the point \(W\). Find the ratio \(AW : WM\). [5]

A tetrahedron \(ABCD\) is such that \(AB\) is perpendicular to the base \(BCD\). 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 \(BCD\) is \(x + 2y - 2z = -1\).

1. Find, in either order, the coordinates of \(B\) and the length of \(AB\). [5]
2. Find the acute angle between the planes \(ACD\) and \(BCD\). [6]

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

1. Obtain the equation of the face \(ABC\) in the form $$x + \sqrt{3}y + \left(\frac{1}{2}\sqrt{2}\right)z = \frac{2}{3}\sqrt{3}.$$ [5] (Answers which only verify the given equation will not receive full credit.)
2. Give a geometrical reason why the equation of the face \(ABD\) can be expressed as $$x - \sqrt{3}y + \left(\frac{1}{2}\sqrt{2}\right)z = \frac{2}{3}\sqrt{3}.$$ [2]
3. Hence find the cosine of the angle between two faces of the tetrahedron. [4]

With respect to an origin \(O\), the points \(A, B, C, D\) have position vectors $$\mathbf{2i - j + k}, \quad \mathbf{i - 2k}, \quad \mathbf{-i + 3j + 2k}, \quad \mathbf{-i + j + 4k},$$ respectively. Find

1. a vector perpendicular to the plane \(OAB\), [2]
2. the acute angle between the planes \(OAB\) and \(OCD\), correct to the nearest \(0.1°\), [3]
3. the shortest distance between the line which passes through \(A\) and \(B\) and the line which passes through \(C\) and \(D\), [4]
4. the perpendicular distance from the point \(A\) to the line which passes through \(C\) and \(D\). [3]