3D geometry applications

9 The quadrilateral \(A B C D\) is a trapezium in which \(A B\) and \(D C\) are parallel. With respect to the origin \(O\), the position vectors of \(A , B\) and \(C\) are given by \(\overrightarrow { O A } = - \mathbf { i } + 2 \mathbf { j } + 3 \mathbf { k } , \overrightarrow { O B } = \mathbf { i } + 3 \mathbf { j } + \mathbf { k }\) and \(\overrightarrow { O C } = 2 \mathbf { i } + 2 \mathbf { j } - 3 \mathbf { k }\).

1. Given that \(\overrightarrow { D C } = 3 \overrightarrow { A B }\), find the position vector of \(D\).
2. State a vector equation for the line through \(A\) and \(B\).
3. Find the distance between the parallel sides and hence find the area of the trapezium.

7 A straight pipeline AB passes through a mountain. With respect to axes \(\mathrm { O } x y z\), with \(\mathrm { O } x\) due East, \(\mathrm { O } y\) due North and \(\mathrm { O } z\) vertically upwards, A has coordinates \(( - 200,100,0 )\) and B has coordinates \(( 100,200,100 )\), where units are metres.

1. Verify that \(\overrightarrow { \mathrm { AB } } = \left( \begin{array} { l } 300 \\ 100 \\ 100 \end{array} \right)\) and find the length of the pipeline.
2. Write down a vector equation of the line AB , and calculate the angle it makes with the vertical. A thin flat layer of hard rock runs through the mountain. The equation of the plane containing this layer is \(x + 2 y + 3 z = 320\).
3. Find the coordinates of the point where the pipeline meets the layer of rock.
4. By calculating the angle between the line AB and the normal to the plane of the layer, find the angle at which the pipeline cuts through the layer.

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}

9 Beside a major route into a county town the authorities decide to build a large pyramid. Fig. 9.1 shows this pyramid, ABCDE O is the centre point of the horizontal base BCDE . A coordinate system is defined with O as the origin. The \(x\)-axis and \(y\)-axis are horizontal and the \(z\)-axis is vertical, as shown in Fig. 9.1 The vertices of the pyramid are $$A ( 0,0,6 ) , B ( - 4 , - 4,0 ) , C ( 4 , - 4,0 ) , D ( 4,4,0 ) \text { and } E ( - 4,4,0 ) .$$ \begin{figure}[h] \end{figure} The pyramid is supported by a vertical pole OA and there are also support rods from O to points on the triangular faces \(\mathrm { ABC } , \mathrm { ACD } , \mathrm { ADE }\) and AEB . One of the rods, ON , is shown in fig.9.2 which shows one quarter of the pyramid. \begin{figure}[h] \end{figure} M is the mid-point of the line BC .

1. Write down the coordinates of M.
2. Write down the vector \(\overrightarrow { \mathrm { AM } }\) and hence the coordinates of the point N which divides \(\overrightarrow { \mathrm { AM } }\) so that the ratio \(\mathrm { AN } : \mathrm { NM } = 2 : 1\).
3. Show that ON is perpendicular to both AM and BC .
4. Hence write down the equation of the plane ABC in its simplest form.
5. Find the angle between the face ABC and the ground.

3 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 \(\mathrm { F } . \mathrm { Q }\) 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.

4 A computer-controlled machine can be programmed to make ats 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 \(20 \mathrm {~cm} \times 30 \mathrm {~cm} \times 30 \mathrm {~cm}\) cuboid is to be at and drilled. The cuboid is positioned relative to \(x\)-, \(y ^ { 2 }\) and z-axes as shown in Fig. 8.1. \begin{figure}[h] \end{figure} \begin{figure}[h] \end{figure} Fkst, a plane out is made to remove the comer at \(E\). The cut goes through the points \(P . Q\) and \(R\), which are the midpoints of the sides \(\mathrm { ED } , \mathrm { EA }\) and EF respectively.

1. Write down the coordinates of \(\mathbf { P } , \mathbf { Q }\) and \(\mathbf { R }\). $$\text { Henceshowlhat } \mathbb { N } ^ { 1 } \left\{ \begin{array} { l } 1 \\ 0 \\ 0 \end{array} \right\} \text { and } \left\{ \begin{array} { l } 1 \\ 0 \end{array} \right\}$$ (;) Show that the veeto, \(\binom { \text { I } } { \text { ) } }\) is pc,pcndicula, to the plone through \(\mathrm { P } , \mathrm { Q }\), nd R
   Hence find the Cartesian equation of this plane. A hole is then drilled perpendicular to lriangle \(P Q R\), as shown in Fig. 82. The hole passes through the triangle at the point \(T\) which divides the line \(P S\) in the ratio 2 : \(I\), where \(S\) is the midpoint of \(Q R\).
2. Write down the coordinates of S , and show that the point T has coordinates \(( - 5,16,25 )\).
3. Write down a vector equation of the line of the drill hole. Hence determine whether or not this line passes through C .

1 Fig. 6 shows a lean-to greenhouse ABCDHEFG . With respect to coordinate axes \(\mathrm { O } x y z\), 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 \(\mathrm { F } . \mathrm { Q }\) 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.

2 Fig. 7 shows a tetrahedron ABCD . The coordinates of the vertices, with respect to axes Oxyz , are \(\mathrm { A } ( - 3,0,0 ) , \mathrm { B } ( 2,0 , - 2 ) , \mathrm { C } ( 0,4,0 )\) and \(\mathrm { D } ( 0,4,5 )\). \begin{figure}[h] \end{figure}

1. Find the length of the edges AB and AC , and the size of the angle CAB . Hence calculate the area of triangle ABC .
2. (A) Verify that \(4 \mathbf { i } - 3 \mathbf { j } + 10 \mathbf { k }\) is normal to the plane ABC .
   (B) Hence find the equation of this plane.
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 . The volume of a tetrahedron is \(\frac { 1 } { 3 } \times\) area of base × height.
4. Find the volume of the tetrahedron ABCD .
5. Find a vector equation of the line \(l\) joining the points \(( 0,1,3 )\) and \(( - 2,2,5 )\).
6. Find the point of intersection of the line \(l\) with the plane \(x + 3 y + 2 z = 4\).
7. Find the acute angle between the line \(l\) and the normal to the plane.

2 A piece of cloth ABDC is attached to the tops of vertical poles \(\mathrm { AE } , \mathrm { BF } , \mathrm { DG }\) and CH , where \(\mathrm { E } , \mathrm { F } , \mathrm { G }\) 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. \begin{figure}[h] \end{figure}

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

3 A straight pipeline AB passes through a mountain. With respect to axes \(\mathrm { O } x y z\), with \(\mathrm { O } x\) due East, Oy due North and \(\mathrm { O } z\) vertically upwards, A has coordinates \(( - 200,100,0 )\) and B has coordinates \(( 100,200,100 )\), where units are metres.

1. Verify that \(\left. \overrightarrow { \mathrm { AB } } = \begin{array} { l } 300 \\ 100 \\ 100 \end{array} \right)\) and find the length of the pipeline.
   [0pt] [3]
2. Write down a vector equation of the line AB , and calculate the angle it makes with the vertical.
   [0pt] [6]
   A thin flat layer of hard rock runs through the mountain. The equation of the plane containing this layer is \(x + 2 y + 3 z = 320\).
3. Find the coordinates of the point where the pipeline meets the layer of rock.
4. By calculating the angle between the line AB and the normal to the plane of the layer, find the angle at which the pipeline cuts through the layer.

4 When a light ray passes from air to glass, it is deflected through an angle. The light ray ABC starts at point \(\mathrm { A } ( 1,2,2 )\), and enters a glass object at point \(\mathrm { B } ( 0,0,2 )\). The surface of the glass object is a plane with normal vector \(\mathbf { n }\). Fig. 7 shows a cross-section of the glass object in the plane of the light ray and \(\mathbf { n }\). \begin{figure}[h] \end{figure}

1. Find the vector \(\overrightarrow { \mathrm { AB } }\) and a vector equation of the line AB . The surface of the glass object is a plane with equation \(x + z = 2\). AB makes an acute angle \(\theta\) with the normal to this plane.
2. Write down the normal vector \(\mathbf { n }\), and hence calculate \(\theta\), giving your answer in degrees. The line BC has vector equation \(\mathbf { r } = \left( \begin{array} { l } 0 \\ 0 \\ 2 \end{array} \right) + \mu \left( \begin{array} { l } - 2 \\ - 2 \\ - 1 \end{array} \right)\). This line makes an acute angle \(\phi\) with the
   normal to the plane. normal to the plane.
3. Show that \(\phi = 45 ^ { \circ }\).
4. Snell's Law states that \(\sin \theta = k \sin \phi\), where \(k\) is a constant called the refractive index. Find \(k\). The light ray leaves the glass object through a plane with equation \(x + z = - 1\). Units are centimetres.
5. Find the point of intersection of the line BC with the plane \(x + z = - 1\). Hence find the distance the light ray travels through the glass object.

1 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.

4 A computer-controlled machine can be programmed to make cuts by entering the equation of the plane of the cut, and to drill boles by entering the equation of the line of the hole. A \(20 \mathrm {~cm} \times 30 \mathrm {~cm} \times 30 \mathrm {~cm}\) cuboid is to be cut and drilled. The cuboid is positioned relative to \(x - y ^ { 2 }\) and \(z\)-axes as shown in Fig. 8.1. \begin{figure}[h] \end{figure} \begin{figure}[h] \end{figure} First, a plane cut is made to remove the comer at \(E\). The cut goes through the points \(P , Q\) and \(R\), which are the midpoints of the sides \(\mathrm { ED } , \mathrm { EA }\) and EF respectively.

1. Write down the coordinates \(\boldsymbol { 0 } \mathbf { F } \mathrm { Q }\) and \(\mathrm { R } \left( \begin{array} { l } F \\ 1 \end{array} \right]\)
   Hence show that \(\mathrm { PQ } = { } _ { - }\): and \(\mathrm { PR } =\)
   (U) Show th,i tho \(, 0010,11\) is pc,pondio,la, to the pl'ute through \(P , Q\) rudd \(R\) Hence find the cartesian equation of this plane. A hole is then drilled perpendicular to triangle PQR , as shown in Fig. 82. The hole passes through the triangle at the point T which divides the line PS in the ratio 2 : I , where S is the midpoint of QR .
2. Write down the coordinates of S , and show that the point T has coordinates \(( - 5.16 \mathrm { i } , 25 )\).
3. Write down a vector equation of the line of the drill hole. Hence determine whether or not this line passes through C .

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 .

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 .

7 A piece of cloth ABDC is attached to the tops of vertical poles \(\mathrm { AE } , \mathrm { BF } , \mathrm { DG }\) and CH , where \(\mathrm { E } , \mathrm { F } , \mathrm { G }\) 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. \begin{figure}[h] \end{figure}

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

7 Fig. 7 shows a tetrahedron ABCD . The coordinates of the vertices, with respect to axes \(\mathrm { O } x y z\), are \(\mathrm { A } ( - 3,0,0 ) , \mathrm { B } ( 2,0 , - 2 ) , \mathrm { C } ( 0,4,0 )\) and \(\mathrm { D } ( 0,4,5 )\). \begin{figure}[h] \end{figure}

1. Find the lengths of the edges AB and AC , and the size of the angle CAB . Hence calculate the area of triangle ABC .
2. (A) Verify that \(4 \mathbf { i } - 3 \mathbf { j } + 10 \mathbf { k }\) is normal to the plane ABC .
   (B) Hence find the equation of this plane.
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 . The volume of a tetrahedron is \(\frac { 1 } { 3 } \times\) area of base × height.
4. Find the volume of the tetrahedron ABCD .

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.}

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.

5. \begin{figure}[h] \end{figure} The points \(A ( 3,2 , - 4 ) , B ( 9 , - 4,2 ) , C ( - 6 , - 10,8 )\) and \(D ( - 4 , - 5,10 )\) are the vertices of a tetrahedron. The plane with equation \(z = 0\) cuts the tetrahedron into two pieces, one on each side of the plane. The edges \(A B , A C\) and \(A D\) of the tetrahedron intersect the plane at the points \(M , N\) and \(P\) respectively, as shown in Figure 1. Determine

1. the coordinates of the points \(M , N\) and \(P\),
2. the area of triangle \(M N P\),
3. the exact volume of the solid \(B C D P N M\).

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.