4.04d Angles: between planes and between line and plane

7 The points \(A , B , C\) have position vectors $$2 \mathbf { i } + 2 \mathbf { j } , \quad - \mathbf { j } + \mathbf { k } \quad \text { and } \quad 2 \mathbf { i } + \mathbf { j } - 7 \mathbf { k }$$ respectively, relative to the origin \(O\).

1. Find an equation of the plane \(O A B\), giving your answer in the form \(\mathbf { r } . \mathbf { n } = p\).
   The plane \(\Pi\) has equation \(\mathrm { x } - 3 \mathrm { y } - 2 \mathrm { z } = 1\).
2. Find the perpendicular distance of \(\Pi\) from the origin.
3. Find the acute angle between the planes \(O A B\) and \(\Pi\).
4. Find an equation for the common perpendicular to the lines \(O C\) and \(A B\).
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

4 The plane \(\Pi\) contains the lines \(\mathbf { r } = 3 \mathbf { i } - 2 \mathbf { j } + \mathbf { k } + \lambda ( - \mathbf { i } + 2 \mathbf { j } + \mathbf { k } )\) and \(\mathbf { r } = 4 \mathbf { i } + 4 \mathbf { j } + 2 \mathbf { k } + \mu ( 3 \mathbf { i } + 2 \mathbf { j } - \mathbf { k } )\).

1. Find a Cartesian equation of \(\Pi\), giving your answer in the form \(a x + b y + c z = d\).
   The line \(l\) passes through the point \(P\) with position vector \(2 \mathbf { i } + 3 \mathbf { j } + \mathbf { k }\) and is parallel to the vector \(\mathbf { j } + \mathbf { k }\).
2. Find the acute angle between \(I\) and \(\Pi\).
3. Find the position vector of the foot of the perpendicular from \(P\) to \(\Pi\).

6 The lines \(l _ { 1 }\) and \(l _ { 2 }\) have equations \(\mathbf { r } = 2 \mathbf { i } + \mathbf { k } + \lambda ( \mathbf { i } - \mathbf { j } + 2 \mathbf { k } )\) and \(\mathbf { r } = 2 \mathbf { j } + 6 \mathbf { k } + \mu ( \mathbf { i } + 2 \mathbf { j } - 2 \mathbf { k } )\) respectively. The point \(P\) on \(l _ { 1 }\) and the point \(Q\) on \(l _ { 2 }\) are such that \(P Q\) is perpendicular to both \(l _ { 1 }\) and \(l _ { 2 }\).

1. Find the length \(P Q\).
   The plane \(\Pi _ { 1 }\) contains \(P Q\) and \(l _ { 1 }\).
   The plane \(\Pi _ { 2 }\) contains \(P Q\) and \(l _ { 2 }\).
2. 1. Write down an equation of \(\Pi _ { 1 }\), giving your answer in the form \(\mathbf { r } = \mathbf { a } + \mathbf { s b } + \mathbf { t c }\).
   2. Find an equation of \(\Pi _ { 2 }\), giving your answer in the form \(a x + b y + c z = d\).
3. Find the acute angle between \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\).

5 The plane \(\Pi _ { 1 }\) has equation \(\mathbf { r } = \mathbf { i } - \mathbf { j } - 2 \mathbf { k } + \lambda ( \mathbf { i } - 2 \mathbf { j } - 3 \mathbf { k } ) + \mu ( 3 \mathbf { i } - \mathbf { k } )\).

1. Find an equation for \(\Pi _ { 1 }\) in the form \(\mathrm { ax } + \mathrm { by } + \mathrm { cz } = \mathrm { d }\).
   The line \(l\), which does not lie in \(\Pi _ { 1 }\), has equation \(\mathbf { r } = - 3 \mathbf { i } + \mathbf { k } + t ( \mathbf { i } + \mathbf { j } + \mathbf { k } )\).
2. Show that \(l\) is parallel to \(\Pi _ { 1 }\).
3. Find the distance between \(l\) and \(\Pi _ { 1 }\).
4. The plane \(\Pi _ { 2 }\) has equation \(3 x + 3 y + 2 z = 1\). Find a vector equation of the line of intersection of \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\).

2 The line \(l _ { 1 }\) has equation \(\mathbf { r } = \mathbf { i } + 3 \mathbf { j } - \mathbf { k } + \lambda ( \mathbf { i } - \mathbf { j } - 4 \mathbf { k } )\).
The plane \(\Pi\) contains \(l _ { 1 }\) and is parallel to the vector \(2 \mathbf { i } + 5 \mathbf { j } - 4 \mathbf { k }\).

1. Find the equation of \(\Pi\), giving your answer in the form \(a x + b y + c z = d\). \includegraphics[max width=\textwidth, alt={}, center]{beb9c1f1-1676-4432-a42a-c418ff9f45d8-05_2723_33_99_22} The line \(l _ { 2 }\) is parallel to the vector \(5 \mathbf { i } - 5 \mathbf { j } - 2 \mathbf { k }\).
2. Find the acute angle between \(l _ { 2 }\) and \(\Pi\).

6 The position vectors of the points \(A , B , C , D\) are $$2 \mathbf { i } + 4 \mathbf { j } - 3 \mathbf { k } , \quad - 2 \mathbf { i } + 5 \mathbf { j } - 4 \mathbf { k } , \quad \mathbf { i } + 4 \mathbf { j } + \mathbf { k } , \quad \mathbf { i } + 5 \mathbf { j } + m \mathbf { k } ,$$ respectively, where \(m\) is an integer. It is given that the shortest distance between the line through \(A\) and \(B\) and the line through \(C\) and \(D\) is 3 .

1. Show that the only possible value of \(m\) is 2 .
2. Find the shortest distance of \(D\) from the line through \(A\) and \(C\).
3. Show that the acute angle between the planes \(A C D\) and \(B C D\) is \(\cos ^ { - 1 } \left( \frac { 1 } { \sqrt { 3 } } \right)\).

1. The plane \(\Pi _ { 1 }\) has vector equation

$$\mathbf { r } . ( 3 \mathbf { i } - 4 \mathbf { j } + 2 \mathbf { k } ) = 5$$

1. Find the perpendicular distance from the point \(( 6,2,12 )\) to the plane \(\Pi _ { 1 }\) The plane \(\Pi _ { 2 }\) has vector equation $$\mathbf { r } = \lambda ( 2 \mathbf { i } + \mathbf { j } + 5 \mathbf { k } ) + \mu ( \mathbf { i } - \mathbf { j } - 2 \mathbf { k } ) , \text { where } \lambda \text { and } \mu \text { are scalar parameters. }$$
2. Find the acute angle between \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\) giving your answer to the nearest degree.
3. Find an equation of the line of intersection of the two planes in the form \(\mathbf { r } \times \mathbf { a } = \mathbf { b }\), where \(\mathbf { a }\) and \(\mathbf { b }\) are constant vectors.

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 \(\mathbf { r } . \mathbf { n } = p\)

3 The planes \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\) have equations \(\mathbf { r } \cdot ( \mathbf { i } - 2 \mathbf { j } + 2 \mathbf { k } ) = 1\) and \(\mathbf { r } \cdot ( 2 \mathbf { i } + 2 \mathbf { j } - \mathbf { k } ) = 3\) respectively. Find

1. the acute angle between \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\), correct to the nearest degree,
2. the equation of the line of intersection of \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\), in the form \(\mathbf { r } = \mathbf { a } + t \mathbf { b }\).

9 A laser beam is aimed from a point ( \(12,10,10\) ) in the direction \(- 2 \mathbf { i } - 2 \mathbf { j } - 3 \mathbf { k }\) towards a plane surface.

1. Give the equation of the path of the laser beam in vector form. The points \(\mathrm { A } ( 1,1,1 ) , \mathrm { B } ( 1,4,2 )\) and \(\mathrm { C } ( 6,1,0 )\) lie on the plane.
2. Show that the vector \(3 \mathbf { i } - 5 \mathbf { j } + 15 \mathbf { k }\) is perpendicular to the plane and hence find the cartesian equation of the plane.
3. Find the coordinate of the point where the laser beam hits the surface of the plane.
4. Find the angle between the laser beam and the plane. \section*{Insert for question 6.} The graph of \(y = \tan x\) is given below.
   On this graph sketch the graph of \(y = \cot x\).
   Show clearly where your graph crosses the graph of \(y = \tan x\) and indicate the asymptotes. [4] \includegraphics[max width=\textwidth, alt={}, center]{23771896-942c-4a1d-ab95-6b6d3cc5643c-5_853_1555_703_262}

2 The points \(\mathrm { A } , \mathrm { B }\) and C have coordinates \(( 1,3 , - 2 ) , ( - 1,2 , - 3 )\) and \(( 0 , - 8,1 )\) respectively.

1. Find the vectors \(\overrightarrow { \mathrm { AB } }\) and \(\overrightarrow { \mathrm { AC } }\).
2. Show that the vector \(2 \mathbf { i } - \mathbf { j } - 3 \mathbf { k }\) is perpendicular to the plane ABC . Hence find the equation of the plane ABC .
3. Write down normal vectors to the planes \(2 x - y + z = 2\) and \(x - z = 1\). Hence find the acute angle between the planes.
4. 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.
5. Find the cartesian equation of the plane through the point \(( 2 , - 1,4 )\) with normal vector $$\mathbf { n } = \left( \begin{array} { l } 1 \\ 1 \\ 2 \end{array} \right) .$$
6. Find the coordinates of the point of intersection of this plane and the straight line with equation $$\mathbf { r } = \left( \begin{array} { r } 7 \\ 12 \\ 9 \end{array} \right) + \lambda \left( \begin{array} { l } 1 \\ 3 \\ 2 \end{array} \right)$$

\begin{figure}[h] \end{figure} Fig. 7 illustrates a house. All units are in metres. The coordinates of A, B, C and E are as shown. BD is horizontal and parallel to AE .

1. Find the length AE .
2. Find a vector equation of the line BD . Given that the length of BD is 15 metres, find the coordinates of D.
3. Verify that the equation of the plane ABC is $$- 3 x + 4 y + 5 z = 30 .$$ Write down a vector normal to this plane.
4. Show that the vector \(\left( \begin{array} { l } 4 \\ 3 \\ 5 \end{array} \right)\) is normal to the plane ABDE . Hence find the equation of the plane ABDE .
5. Find the angle between the planes ABC and ABDE .

4. \begin{figure}[h] \end{figure} Figure 1 shows a cuboid \(O A B C D E F G\), where \(O\) is the origin, \(A\) has position vector \(5 \mathbf { i } , C\) has position vector \(10 \mathbf { j }\) and \(D\) has position vector \(20 \mathbf { k }\).

1. Find the cosine of angle \(C A F\). Given that the point \(X\) lies on \(A C\) and that \(F X\) is perpendicular to \(A C\),
2. find the position vector of point \(X\) and the distance \(F X\). The line \(l _ { 1 }\) passes through \(O\) and through the midpoint of the face \(A B F E\). The line \(l _ { 2 }\) passes through \(A\) and through the midpoint of the edge \(F G\).
3. Show that \(l _ { 1 }\) and \(l _ { 2 }\) intersect and find the coordinates of the point of intersection.

6
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 \(\mathbf{r} \cdot \mathbf{n} = p\).
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 \(A W : W M\).

1 Two planes have equations $$x + 2 y + 5 z = 12 \text { and } 2 x - y + 3 z = 5$$

1. Find the acute angle between the planes.
2. Find a vector equation of the line of intersection of the planes.

1 The plane \(p\) has equation \(\mathbf { r } . ( \mathbf { i } - 3 \mathbf { j } + 4 \mathbf { k } ) = 4\) and the line \(l _ { 1 }\) has equation \(\mathbf { r } = 2 \mathbf { j } - \mathbf { k } + t ( 3 \mathbf { i } + \mathbf { j } + 2 \mathbf { k } )\). The line \(l _ { 2 }\) is parallel to \(p\) and perpendicular to \(l _ { 1 }\), and passes through the point with position vector \(\mathbf { i } + 4 \mathbf { j } + 2 \mathbf { k }\). Find the equation of \(l _ { 2 }\), giving your answer in the form \(\mathbf { r } = \mathbf { a } + t \mathbf { b }\).

6 The plane \(\Pi\) has equation \(x + 2 y - 2 z = 5\). The line \(l\) has equation \(\frac { x - 1 } { 2 } = \frac { y + 1 } { 5 } = \frac { z - 2 } { 1 }\).

1. Find the coordinates of the point of intersection of \(l\) with the plane \(\Pi\).
2. Calculate the acute angle between \(l\) and \(\Pi\).
3. Find the coordinates of the two points on the line \(l\) such that the distance of each point from the plane \(\Pi\) is 2 .

3 The plane \(\Pi\) passes through the points \(( 1,2,1 ) , ( 2,3,6 )\) and \(( 4 , - 1,2 )\).

1. Find a cartesian equation of the plane \(\Pi\). The line \(l\) has equation \(\mathbf { r } = \left( \begin{array} { r } - 1 \\ - 2 \\ 6 \end{array} \right) + \lambda \left( \begin{array} { r } 4 \\ 3 \\ - 2 \end{array} \right)\).
2. Find the coordinates of the point of intersection of \(\Pi\) and \(l\).
3. Find the acute angle between \(\Pi\) and \(l\).

5

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.

7 A tent has vertices ABCDEF with coordinates as shown in Fig. 7. Lengths are in metres. The \(\mathrm { O } x y\) plane is horizontal. \begin{figure}[h] \end{figure}

1. Find the length of the ridge of the tent DE , and the angle this makes with the horizontal.
2. Show that the vector \(\mathbf { i } - 4 \mathbf { j } + 5 \mathbf { k }\) is normal to the plane through \(\mathrm { A } , \mathrm { D }\) and E . Hence find the equation of this plane. Given that B lies in this plane, find \(a\).
3. Verify that the equation of the plane BCD is \(x + z = 8\). Hence find the acute angle between the planes ABDE and BCD .

6 Fig. 6 shows a lean-to greenhouse ABCDHEFG . With respect to coordinate axes Oxyz , the coordinates of the vertices are as shown. All distances are in metres. Ground level is the plane \(z = 0\). \begin{figure}[h] \end{figure}

1. Verify that the equation of the plane through \(\mathrm { A } , \mathrm { B }\) and E is \(x + 6 y + 12 = 0\). Hence, given that F lies in this plane, show that \(a = - 2 \frac { 1 } { 3 }\).
2. (A) Show that the vector \(\left( \begin{array} { r } 1 \\ - 6 \\ 0 \end{array} \right)\) is normal to the plane DHC.
   (B) Hence find the cartesian equation of this plane.
   (C) Given that G lies in the plane DHC , find \(b\) and the length FG .
3. Find the angle EFB . A straight wire joins point H to a point P which is half way between E and F . Q is a point two-thirds of the way along this wire, so that \(\mathrm { HQ } = 2 \mathrm { QP }\).
4. Find the height of Q above the ground. \section*{Question 7 begins on page 4.}

10 The lines \(l _ { 1 }\) and \(l _ { 2 }\) have equations $$l _ { 1 } : \mathbf { r } = 6 \mathbf { i } + 5 \mathbf { j } + 4 \mathbf { k } + \lambda ( \mathbf { i } + \mathbf { j } + \mathbf { k } ) \quad \text { and } \quad l _ { 2 } : \mathbf { r } = 6 \mathbf { i } + 5 \mathbf { j } + 4 \mathbf { k } + \mu ( 4 \mathbf { i } + 6 \mathbf { j } + \mathbf { k } ) .$$ Find a cartesian equation of the plane \(\Pi\) containing \(l _ { 1 }\) and \(l _ { 2 }\). Find the position vector of the foot of the perpendicular from the point with position vector \(\mathbf { i } + 10 \mathbf { j } + 3 \mathbf { k }\) to \(\Pi\). The line \(l _ { 3 }\) has equation \(\mathbf { r } = \mathbf { i } + 10 \mathbf { j } + 3 \mathbf { k } + v ( 2 \mathbf { i } - 3 \mathbf { j } + \mathbf { k } )\). Find the shortest distance between \(l _ { 1 }\) and \(l _ { 3 }\).

The position vectors of the points \(A , B , C , D\) are $$2 \mathbf { i } + 4 \mathbf { j } - 3 \mathbf { k } , \quad - 2 \mathbf { i } + 5 \mathbf { j } - 4 \mathbf { k } , \quad \mathbf { i } + 4 \mathbf { j } + \mathbf { k } , \quad \mathbf { i } + 5 \mathbf { j } + m \mathbf { k }$$ respectively, where \(m\) is an integer. It is given that the shortest distance between the line through \(A\) and \(B\) and the line through \(C\) and \(D\) is 3 . Show that the only possible value of \(m\) is 2 . Find the shortest distance of \(D\) from the line through \(A\) and \(C\). Show that the acute angle between the planes \(A C D\) and \(B C D\) is \(\cos ^ { - 1 } \left( \frac { 1 } { \sqrt { } 3 } \right)\).

The position vectors of the points \(A , B , C , D\) are $$2 \mathbf { i } + 4 \mathbf { j } - 3 \mathbf { k } , \quad - 2 \mathbf { i } + 5 \mathbf { j } - 4 \mathbf { k } , \quad \mathbf { i } + 4 \mathbf { j } + \mathbf { k } , \quad \mathbf { i } + 5 \mathbf { j } + m \mathbf { k } ,$$ respectively, where \(m\) is an integer. It is given that the shortest distance between the line through \(A\) and \(B\) and the line through \(C\) and \(D\) is 3 . Show that the only possible value of \(m\) is 2 . Find the shortest distance of \(D\) from the line through \(A\) and \(C\). Show that the acute angle between the planes \(A C D\) and \(B C D\) is \(\cos ^ { - 1 } \left( \frac { 1 } { \sqrt { } 3 } \right)\).

9 The plane \(\Pi _ { 1 }\) has parametric equation $$\mathbf { r } = 2 \mathbf { i } - 3 \mathbf { j } + \mathbf { k } + \lambda ( \mathbf { i } - 2 \mathbf { j } - \mathbf { k } ) + \mu ( \mathbf { i } + 2 \mathbf { j } - 2 \mathbf { k } )$$ Find a cartesian equation of \(\Pi _ { 1 }\). The plane \(\Pi _ { 2 }\) has cartesian equation \(3 x - 2 y - 3 z = 4\). Find the acute angle between \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\). Find a vector equation of the line of intersection of \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\).