4.04d Angles: between planes and between line and plane

13 The points A and B have coordinates \(( 4,0 , - 1 )\) and \(( 10,4 , - 3 )\) respectively. The planes \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\) have equations \(x - 2 y = 5\) and \(2 x + 3 y - z = - 4\) respectively.

1. Find the acute angle between the line AB and the plane \(\Pi _ { 1 }\).
2. Show that the line AB meets \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\) at the same point, whose coordinates should be specified.
3. 1. Find \(( \mathbf { i } - 2 \mathbf { j } ) \times ( 2 \mathbf { i } + 3 \mathbf { j } - \mathbf { k } )\).
   2. Hence find the acute angle between the planes \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\).
   3. Find the shortest distance between the point A and the line of intersection of the planes \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\).

2 In this question you must show detailed reasoning.
Find the angle between the vector \(3 i + 2 j + \mathbf { k }\) and the plane \(- x + 3 y + 2 z = 8\).

11 The plane \(\Pi\) has equation \(2 x - y + 2 z = 4\). The point \(P\) has coordinates \(( 8,4,5 )\).

1. Calculate the shortest distance from P to \(\Pi\). The line \(L\) has equation \(\frac { x - 2 } { 3 } = \frac { y } { 2 } = \frac { z + 3 } { 4 }\).
2. Verify that P lies on L .
3. Find the coordinates of the point of intersection of L and \(\Pi\).
4. Determine the acute angle between L and \(\Pi\).
5. Use the results of parts (b), (c) and (d) to verify your answer to part (a).

7. The vector equations of the lines \(L _ { 1 } , L _ { 2 } , L _ { 3 }\) are given by $$\begin{aligned} & \mathbf { r } = 3 \mathbf { i } + 2 \mathbf { j } + \mathbf { k } + \lambda ( 2 \mathbf { i } + n \mathbf { j } + \mathbf { k } ) \\ & \mathbf { r } = 5 \mathbf { i } - 3 \mathbf { j } - 4 \mathbf { k } + \mu ( 3 \mathbf { i } + \mathbf { j } - 3 \mathbf { k } ) \\ & \mathbf { r } = 6 \mathbf { i } - 3 \mathbf { j } + 2 \mathbf { k } + v ( p \mathbf { i } + 3 \mathbf { j } + 4 \mathbf { k } ) \end{aligned}$$ respectively, where \(n\) and \(p\) are constants.
The line \(L _ { 1 }\) is perpendicular to the line \(L _ { 2 }\). The line \(L _ { 1 }\) is also perpendicular to the line \(L _ { 3 }\).

1. Show that the value of \(n\) is - 3 and find the value of \(p\).
2. Find the acute angle between the lines \(L _ { 2 }\) and \(L _ { 3 }\).

1. A mining company has identified a mineral layer below ground.

The mining company wishes to drill down to reach the mineral layer and models the situation as follows. With respect to a fixed origin \(O\),

• the ground is modelled as a horizontal plane with equation \(z = 0\)
• the mineral layer is modelled as part of the plane containing the points \(A ( 10,5 , - 50 ) , B ( 15,30 , - 45 )\) and \(C ( - 5,20 , - 60 )\), where the units are in metres
  1. Determine an equation for the plane containing \(A , B\) and \(C\), giving your answer in the form r.n \(= d\)
  2. Determine, according to the model, the acute angle between the ground and the plane containing the mineral layer. Give your answer to the nearest degree.

The mining company plans to drill vertically downwards from the point \(( 5,12,0 )\) on the ground to reach the mineral layer.

Using the model, determine, in metres to 1 decimal place, the distance the mining company will need to drill in order to reach the mineral layer.

State a limitation of the assumption that the mineral layer can be modelled as a plane.

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

$$\mathbf { r } = 2 \mathbf { i } + 4 \mathbf { j } - \mathbf { k } + \lambda ( \mathbf { i } + 2 \mathbf { j } - 3 \mathbf { k } ) + \mu ( - \mathbf { i } + 2 \mathbf { j } + \mathbf { k } )$$ where \(\lambda\) and \(\mu\) are scalar parameters.

1. Find a Cartesian equation for \(\Pi _ { 1 }\) The line \(l\) has equation $$\frac { x - 1 } { 5 } = \frac { y - 3 } { - 3 } = \frac { z + 2 } { 4 }$$
2. Find the coordinates of the point of intersection of \(l\) with \(\Pi _ { 1 }\) The plane \(\Pi _ { 2 }\) has equation $$\mathbf { r . } ( 2 \mathbf { i } - \mathbf { j } + 3 \mathbf { k } ) = 5$$
3. Find, to the nearest degree, the acute angle between \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\)

1. The line \(l _ { 1 }\) has equation

$$\mathbf { r } = \mathbf { i } - 2 \mathbf { j } + 3 \mathbf { k } + \lambda ( 2 \mathbf { i } + \mathbf { j } - 4 \mathbf { k } )$$ and the line \(l _ { 2 }\) has equation $$\mathbf { r } = 5 \mathbf { i } + p \mathbf { j } - 7 \mathbf { k } + \mu ( 6 \mathbf { i } + \mathbf { j } + 8 \mathbf { k } )$$ where \(\lambda\) and \(\mu\) are scalar parameters and \(p\) is a constant.
The plane \(\Pi\) contains \(l _ { 1 }\) and \(l _ { 2 }\)

1. Show that the vector \(3 \mathbf { i } - 10 \mathbf { j } - \mathbf { k }\) is perpendicular to \(\Pi\)
2. Hence determine a Cartesian equation of \(\Pi\)
3. Hence determine the value of \(p\) Given that
   • the lines \(l _ { 1 }\) and \(l _ { 2 }\) intersect at the point \(A\)
   • the point \(B\) has coordinates \(( 12 , - 11,6 )\)
   • determine, to the nearest degree, the acute angle between \(A B\) and \(\Pi\)

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

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

1. Find the perpendicular distance from the point \(( 6,2,12 )\) to the plane \(\Pi _ { 1 }\) The plane \(\Pi _ { 2 }\) has vector equation $$\mathbf { r } = \lambda ( 2 \mathbf { i } + \mathbf { j } + 5 \mathbf { k } ) + \mu ( \mathbf { i } - \mathbf { j } - 2 \mathbf { k } )$$ where \(\lambda\) and \(\mu\) are scalar parameters.
2. Show that the vector \(- \mathbf { i } - 3 \mathbf { j } + \mathbf { k }\) is perpendicular to \(\Pi _ { 2 }\)
3. Show that the acute angle between \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\) is \(52 ^ { \circ }\) to the nearest degree.

1. 1. The line \(l _ { 1 }\) has equation \(\mathbf { r } = \left( \begin{array} { r } 2 \\ - 3 \\ 1 \end{array} \right) + \lambda \left( \begin{array} { r } 3 \\ 4 \\ - 1 \end{array} \right)\)

The line \(l _ { 2 }\) has equation \(\mathbf { r } = \left( \begin{array} { c } 13 \\ 5 \\ 8 \end{array} \right) + \mu \left( \begin{array} { r } 1 \\ - 2 \\ 5 \end{array} \right)\) where \(\lambda\) and \(\mu\) are scalar parameters.
The lines \(l _ { 1 }\) and \(l _ { 2 }\) intersect at the point \(P\).

1. Determine the coordinates of \(P\). Given that the plane \(\Pi\) contains both \(l _ { 1 }\) and \(l _ { 2 }\)
2. determine a Cartesian equation for \(\Pi\).
   (ii) Determine a Cartesian equation for each of the two lines that
   • pass through \(( 0,0,0 )\)
   • make an angle of \(60 ^ { \circ }\) with the \(x\)-axis
   • make an angle of \(45 ^ { \circ }\) with the \(y\)-axis

9 In this question you must show detailed reasoning. Find a vector \(\mathbf { v }\) which has the following properties.

• It is a unit vector.
• It is parallel to the plane \(2 x + 2 y + z = 10\).
• It makes an angle of \(45 ^ { \circ }\) with the normal to the plane \(\mathrm { x } + \mathrm { z } = 5\).

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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.

6 The equation of a plane \(\Pi\) is \(x - 2 y - z = 30\).

1. Find the acute angle between the line \(\mathbf { r } = \left( \begin{array} { c } 3 \\ 2 \\ - 5 \end{array} \right) + \lambda \left( \begin{array} { r } - 5 \\ 3 \\ 2 \end{array} \right)\) and \(\Pi\).
2. Determine the geometrical relationship between the line \(\mathbf { r } = \left( \begin{array} { l } 1 \\ 4 \\ 2 \end{array} \right) + \mu \left( \begin{array} { r } 3 \\ - 1 \\ 5 \end{array} \right)\) and \(\Pi\).

8 The upper and lower surfaces of a coal seam are modelled as planes ABC and DEF, as shown in Fig. 8. All dimensions are metres. \begin{figure}[h] \end{figure} Relative to axes \(\mathrm { O } x\) (due east), \(\mathrm { O } y\) (due north) and \(\mathrm { O } z\) (vertically upwards), the coordinates of the points are as follows.
A: (0, 0, -15)
B: (100, 0, -30)
C: (0, 100, -25)
D: (0, 0, -40)
E: (100, 0, -50)
F: (0, 100, -35)

1. Verify that the cartesian equation of the plane ABC is \(3 x + 2 y + 20 z + 300 = 0\).
2. Find the vectors \(\overrightarrow { \mathrm { DE } }\) and \(\overrightarrow { \mathrm { DF } }\). Show that the vector \(2 \mathbf { i } - \mathbf { j } + 20 \mathbf { k }\) is perpendicular to each of these vectors. Hence find the cartesian equation of the plane DEF .
3. By calculating the angle between their normal vectors, find the angle between the planes ABC and DEF. It is decided to drill down to the seam from a point \(\mathrm { R } ( 15,34,0 )\) in a line perpendicular to the upper surface of the seam. This line meets the plane ABC at the point S .
4. Write down a vector equation of the line RS. Calculate the coordinates of S.

The points \(A , B\) and \(C\) have position vectors \(\mathbf { i } , 2 \mathbf { j }\) and \(4 \mathbf { k }\) respectively, relative to an origin \(O\). The point \(N\) is the foot of the perpendicular from \(O\) to the plane \(A B C\). The point \(P\) on the line-segment \(O N\) is such that \(O P = \frac { 3 } { 4 } O N\). The line \(A P\) meets the plane \(O B C\) at \(Q\). Find a vector perpendicular to the plane \(A B C\) and show that the length of \(O N\) is \(\frac { 4 } { \sqrt { } ( 21 ) }\). Find the position vector of the point \(Q\). Show that the acute angle between the planes \(A B C\) and \(A B Q\) is \(\cos ^ { - 1 } \left( \frac { 2 } { 3 } \right)\).

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]

9 Three non-collinear points \(A , B\) and \(C\) have position vectors \(\mathbf { a } , \mathbf { b }\) and \(\mathbf { c }\) respectively, relative to the origin \(O\). The plane through \(A , B\) and \(C\) is denoted by \(\Pi\).

1. (a) Prove that the area of triangle \(A B C\) is \(\frac { 1 } { 2 } | \mathbf { a } \times \mathbf { b } + \mathbf { b } \times \mathbf { c } + \mathbf { c } \times \mathbf { a } |\).
   (b) Describe the significance of the vector \(\mathbf { a } \times \mathbf { b } + \mathbf { b } \times \mathbf { c } + \mathbf { c } \times \mathbf { a }\) in relation to \(\Pi\).
2. (a) In the case when \(\mathbf { a } = a \mathbf { i } , \mathbf { b } = b \mathbf { j }\) and \(\mathbf { c } = c \mathbf { k }\), where \(a , b\) and \(c\) are positive scalar constants, determine the equation of \(\Pi\) in the form r.n \(= d\), where the components of \(\mathbf { n }\) and the value of the scalar constant \(d\) are to be given in terms of \(a , b\) and \(c\).
   (b) Deduce the shortest distance from the origin \(O\) to \(\Pi\) in this case.

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

1. a vector perpendicular to the plane \(O A B\),
2. the acute angle between the planes \(O A B\) and \(O C D\), correct to the nearest \(0.1 ^ { \circ }\),
3. the shortest distance between the line \(A B\) and the line \(C D\),
4. the perpendicular distance from the point \(A\) to the line \(C D\).

The points \(A\) and \(B\) have position vectors, relative to the origin \(O\), given by $$\overrightarrow{OA} = \begin{pmatrix} -1 \\ 3 \\ 5 \end{pmatrix} \quad \text{and} \quad \overrightarrow{OB} = \begin{pmatrix} 3 \\ -1 \\ -4 \end{pmatrix}.$$ The line \(l\) passes through \(A\) and is parallel to \(OB\). The point \(N\) is the foot of the perpendicular from \(B\) to \(l\).

1. State a vector equation for the line \(l\). [1]
2. Find the position vector of \(N\) and show that \(BN = 3\). [6]
3. Find the equation of the plane containing \(A\), \(B\) and \(N\), giving your answer in the form \(ax + by + cz = d\). [5]

The straight line \(l\) has equation \(\mathbf{r} = 2\mathbf{i} - \mathbf{j} - 4\mathbf{k} + \lambda(\mathbf{i} + 2\mathbf{j} + 2\mathbf{k})\). The plane \(p\) has equation \(3x - y + 2z = 9\). The line \(l\) intersects the plane \(p\) at the point \(A\).

1. Find the position vector of \(A\). [3]
2. Find the acute angle between \(l\) and \(p\). [4]
3. Find an equation for the plane which contains \(l\) and is perpendicular to \(p\), giving your answer in the form \(ax + by + cz = d\). [5]

The points \(A\) and \(B\) have position vectors \(\mathbf{2i - 3j + 2k}\) and \(\mathbf{5i - 2j + k}\) respectively. The plane \(p\) has equation \(x + y = 5\).

1. Find the position vector of the point of intersection of the line through \(A\) and \(B\) and the plane \(p\). [4]
2. A second plane \(q\) has an equation of the form \(x + by + cz = d\), where \(b\), \(c\) and \(d\) are constants. The plane \(q\) contains the line \(AB\), and the acute angle between the planes \(p\) and \(q\) is \(60°\). Find the equation of \(q\). [7]

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]

The lines \(l_1\) and \(l_2\) have equations \(\mathbf{r} = \mathbf{i} + 3\mathbf{j} - 2\mathbf{k} + \lambda(2\mathbf{i} + \mathbf{j} + \mathbf{k})\) and \(\mathbf{r} = \mathbf{i} - 2\mathbf{j} + 9\mathbf{k} + \mu(\mathbf{i} - 4\mathbf{j} + 2\mathbf{k})\) respectively. The plane \(\Pi_1\) contains \(l_1\) and is parallel to \(l_2\).

1. Find the equation of \(\Pi_1\), giving your answer in the form \(ax + by + cz = d\). [4]

The plane \(\Pi_2\) contains \(l_2\) and the point with coordinates \((2, -1, 7)\).

1. Find the acute angle between \(\Pi_1\) and \(\Pi_2\). [4]

The point \(P\) on \(l_1\) and the point \(Q\) on \(l_2\) are such that \(PQ\) is perpendicular to both \(l_1\) and \(l_2\).

1. Find a vector equation for \(PQ\). [7]

The line \(l_1\) passes through the point \(A\) with position vector \(\mathbf{i} - \mathbf{j} - 2\mathbf{k}\) and is parallel to the vector \(3\mathbf{i} - 4\mathbf{j} - 2\mathbf{k}\). The variable line \(l_2\) passes through the point \((1 + 5 \cos t)\mathbf{i} - (1 + 5 \sin t)\mathbf{j} - 14\mathbf{k}\), where \(0 \leq t < 2\pi\), and is parallel to the vector \(15\mathbf{i} + 8\mathbf{j} - 3\mathbf{k}\). The points \(P\) and \(Q\) are on \(l_1\) and \(l_2\) respectively, and \(PQ\) is perpendicular to both \(l_1\) and \(l_2\).

1. Find the length of \(PQ\) in terms of \(t\). [4]
2. Hence show that the lines \(l_1\) and \(l_2\) do not intersect, and find the maximum length of \(PQ\) as \(t\) varies. [3]
3. The plane \(\Pi_1\) contains \(l_1\) and \(PQ\); the plane \(\Pi_2\) contains \(l_2\) and \(PQ\). Find the angle between the planes \(\Pi_1\) and \(\Pi_2\), correct to the nearest tenth of a degree. [4]

Answer only one of the following two alternatives. EITHER The points \(A\), \(B\) and \(C\) have position vectors \(\mathbf{i}\), \(2\mathbf{j}\) and \(4\mathbf{k}\) respectively, relative to an origin \(O\). The point \(N\) is the foot of the perpendicular from \(O\) to the plane \(ABC\). The point \(P\) on the line-segment \(ON\) is such that \(OP = \frac{3}{4}ON\). The line \(AP\) meets the plane \(OBC\) at \(Q\). Find a vector perpendicular to the plane \(ABC\) and show that the length of \(ON\) is \(\frac{1}{\sqrt{(21)}}\). [4] Find the position vector of the point \(Q\). [5] Show that the acute angle between the planes \(ABC\) and \(ABQ\) is \(\cos^{-1}\left(\frac{4}{5}\right)\). [5] OR The curve \(C\) has polar equation \(r = a(1 - \cos\theta)\) for \(0 \leqslant \theta < 2\pi\). Sketch \(C\). [2] Find the area of the region enclosed by the arc of \(C\) for which \(\frac{1}{3}\pi \leqslant \theta \leqslant \frac{2}{3}\pi\), the half-line \(\theta = \frac{1}{3}\pi\) and the half-line \(\theta = \frac{2}{3}\pi\). [5] Show that $$\left(\frac{\mathrm{d}s}{\mathrm{d}\theta}\right)^2 = 4a^2\sin^2\left(\frac{1}{2}\theta\right),$$ where \(s\) denotes arc length, and find the length of the arc of \(C\) for which \(\frac{1}{3}\pi \leqslant \theta \leqslant \frac{2}{3}\pi\). [7]

The plane \(\Pi_1\) has equation $$\mathbf{r} = \begin{pmatrix} 5 \\ 1 \\ 0 \end{pmatrix} + s\begin{pmatrix} -4 \\ 1 \\ 3 \end{pmatrix} + t\begin{pmatrix} 0 \\ 1 \\ 2 \end{pmatrix}.$$

1. Find a cartesian equation of \(\Pi_1\). [3]

The plane \(\Pi_2\) has equation \(3x + y - z = 3\).

1. Find the acute angle between \(\Pi_1\) and \(\Pi_2\), giving your answer in degrees. [2]
2. 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]