Line-plane intersection and related angle/perpendicularity

1

1. Find the point of intersection of the line \(\left. \left. \mathbf { r } = \begin{array} { r } - 8 \\ - 2 \\ 6 \end{array} \right) + \lambda \begin{array} { r } - 3 \\ 0 \\ 1 \end{array} \right)\) and the plane \(2 x - 3 y + z = 11\).
2. Find the acute angle between the line and the normal to the plane.

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 drainage system for a sports field consists of underground pipes.

This situation is modelled with respect to a fixed origin \(O\).
According to the model,

• the surface of the sports field is a plane with equation \(z = 0\)
• the pipes are straight lines
• one of the pipes, \(P _ { 1 }\), passes through the points \(A ( 3,4 , - 2 )\) and \(B ( - 2 , - 8 , - 3 )\)
• a different pipe, \(P _ { 2 }\), has equation \(\frac { x - 1 } { 2 } = \frac { y - 3 } { 4 } = \frac { z + 1 } { - 2 }\)
• the units are metres
  1. Determine a vector equation of the line representing the pipe \(P _ { 1 }\)
  2. Determine the coordinates of the point at which the pipe \(P _ { 1 }\) meets the surface of the playing field, according to the model.

Determine, according to the model,

the acute angle between pipes \(P _ { 1 }\) and \(P _ { 2 }\), giving your answer in degrees to 3 significant figures,

the shortest distance between pipes \(P _ { 1 }\) and \(P _ { 2 }\)

5

1. Find the cartesian equation of the plane through the point ( \(2 , - 1,4\) ) with normal vector $$\mathbf { n } = \left( \begin{array} { r } 1 \\ - 1 \\ 2 \end{array} \right)$$
2. Find the coordinates of the point of intersection of this plane and the straight line with equation $$\mathbf { r } = \left( \begin{array} { r } 7 \\ 12 \\ 9 \end{array} \right) + \lambda \left( \begin{array} { l } 1 \\ 3 \\ 2 \end{array} \right)$$

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]

Verify that the vector \(2\mathbf{i} - \mathbf{j} + 4\mathbf{k}\) is perpendicular to the plane through the points A(2, 0, 1), B(1, 2, 2) and C(0, -4, 1). Hence find the cartesian equation of the plane. [5]

A piece of cloth ABDC is attached to the tops of vertical poles AE, BF, DG and CH, where E, F, G and H are at ground level (see Fig. 7). Coordinates are as shown, with lengths in metres. The length of pole DG is \(k\) metres. \includegraphics{figure_7}

1. Write down the vectors \(\overrightarrow{AB}\) and \(\overrightarrow{AC}\). Hence calculate the angle BAC. [6]
2. Verify that the equation of the plane ABC is \(x + y - 2z + d = 0\), where \(d\) is a constant to be determined. Calculate the acute angle the plane makes with the horizontal plane. [7]
3. Given that A, B, D and C are coplanar, show that \(k = 3\). Hence show that ABDC is a trapezium, and find the ratio of CD to AB. [5]

With respect to cartesian coordinates \(Oxyz\), a laser beam ABC is fired from the point A(1, 2, 4), and is reflected at point B off the plane with equation \(x + 2y - 3z = 0\), as shown in Fig. 8. A' is the point (2, 4, 1), and M is the midpoint of AA'. \includegraphics{figure_8}

1. Show that AA' is perpendicular to the plane \(x + 2y - 3z = 0\), and that M lies in the plane. [4]

The vector equation of the line AB is \(\mathbf{r} = \begin{pmatrix} 1 \\ 2 \\ 4 \end{pmatrix} + \lambda \begin{pmatrix} 1 \\ 1 \\ 2 \end{pmatrix}\).

1. Find the coordinates of B, and a vector equation of the line A'B. [6]
2. Given that A'BC is a straight line, find the angle \(\theta\). [4]
3. Find the coordinates of the point where BC crosses the \(Oxz\) plane (the plane containing the \(x\)- and \(z\)-axes). [3]

Fig. 7 shows a tetrahedron ABCD. The coordinates of the vertices, with respect to axes Oxyz, are A(-3, 0, 0), B(2, 0, -2), C(0, 4, 0) and D(0, 4, 5). \includegraphics{figure_7}

1. Find the lengths of the edges AB and AC, and the size of the angle CAB. Hence calculate the area of triangle ABC. [7]
2. 1. Verify that 4i - 3j + 10k is normal to the plane ABC. [2]
   2. Hence find the equation of this plane. [2]
3. Write down a vector equation for the line through D perpendicular to the plane ABC. Hence find the point of intersection of this line with the plane ABC. [5]

The volume of a tetrahedron is \(\frac{1}{3} \times \text{area of base} \times \text{height}\).

1. Find the volume of the tetrahedron ABCD. [2]

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. \includegraphics{figure_1} Relative to axes \(Ox\) (due east), \(Oy\) (due north) and \(Oz\) (vertically upwards), the coordinates of the points are as follows. A: \((0, 0, -15)\) \quad B: \((100, 0, -30)\) \quad C: \((0, 100, -25)\) D: \((0, 0, -40)\) \quad E: \((100, 0, -50)\) \quad F: \((0, 100, -35)\)

1. Verify that the cartesian equation of the plane ABC is \(3x + 2y + 20z + 300 = 0\). [3]
2. Find the vectors \(\overrightarrow{DE}\) and \(\overrightarrow{DF}\). 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. [6]
3. By calculating the angle between their normal vectors, find the angle between the planes ABC and DEF. [4]

It is decided to drill down to the seam from a point 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.

1. Write down a vector equation of the line RS. Find the coordinates of S. [5]

A computer-controlled machine can be programmed to make cuts by entering the equation of the plane of the cut, and to drill holes by entering the equation of the line of the hole. A \(20\text{ cm} \times 30\text{ cm} \times 30\text{ cm}\) cuboid is to be cut and drilled. The cuboid is positioned relative to \(x\)-, \(y\)- and \(z\)-axes as shown in Fig. 8.1. \includegraphics{figure_2} First, a plane cut is made to remove the corner at E. The cut goes through the points P, Q and R, which are the midpoints of the sides ED, EA and EF respectively.

1. Write down the coordinates of P, Q and R. Hence show that \(\overrightarrow{PQ} = \begin{pmatrix} 0 \\ 0 \\ -15 \end{pmatrix}\) and \(\overrightarrow{PR} = \begin{pmatrix} -15 \\ 0 \\ 1 \end{pmatrix}\). [4]
2. Show that \(\begin{pmatrix} 0 \\ 1 \\ 0 \end{pmatrix}\) is perpendicular to the plane through P, Q and R. Hence find the cartesian equation of this plane. [5]

A hole is then drilled perpendicular to triangle PQR, as shown in Fig. 8.2. The hole passes through the triangle at the point T which divides the line PS in the ratio \(2:1\), where S is the midpoint of QR.

1. Write down the coordinates of S, and show that the point T has coordinates \((-5, 16, 25)\). [4]
2. Write down a vector equation of the line of the drill hole. Hence determine whether or not this line passes through C. [5]

A tent has vertices ABCDEF with coordinates as shown in Fig. 7. Lengths are in metres. The \(Oxy\) plane is horizontal. \includegraphics{figure_3}

1. Find the length of the ridge of the tent DE, and the angle this makes with the horizontal. [4]
2. Show that the vector \(\mathbf{i} - 4\mathbf{j} + 5\mathbf{k}\) is normal to the plane through A, D and E. Hence find the equation of this plane. Given that B lies in this plane, find \(a\). [7]
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]

The plane \(\Pi\) has equation \(3x - 5y + z = 9\).

1. Show that \(\Pi\) contains
   and
   [4]
2. Determine the equation of a plane which is perpendicular to \(\Pi\) and which passes through \((4,1,2)\). [3]

The line \(l_1\) has equation \(\mathbf{r} = \begin{pmatrix} 1 \\ -3 \\ 3 \end{pmatrix} + \lambda \begin{pmatrix} 3 \\ 2 \\ -2 \end{pmatrix}\). The plane \(\Pi\) has equation \(\mathbf{r} \cdot \begin{pmatrix} 2 \\ 1 \\ -3 \end{pmatrix} = 4\).

1. Find the position vector of the point of intersection of \(l_1\) and \(\Pi\). [3]
2. Find the acute angle between \(l_1\) and \(\Pi\). [3]

\(A\) is the point on \(l_1\) where \(\lambda = 1\). \(l_2\) is the line with the following properties. • \(l_2\) passes through \(A\) • \(l_2\) is perpendicular to \(l_1\) • \(l_2\) is parallel to \(\Pi\)

1. Find, in vector form, the equation of \(l_2\). [3]

Three planes have equations \begin{align} (4k + 1)x - 3y + (k - 5)z &= 3
(k - 1)x + (3 - k)y + 2z &= 1
7x - 3y + 4z &= 2 \end{align}

1. The planes do not meet at a unique point. Show that \(k = 4.5\) is one possible value of \(k\), and find the other possible value of \(k\). [3 marks]
2. For each value of \(k\) found in part (a), identify the configuration of the given planes. In each case fully justify your answer, stating whether or not the equations of the planes form a consistent system. [4 marks]