Vector geometry in 3D shapes

5 \includegraphics[max width=\textwidth, alt={}, center]{b2cefbd6-6e89-495a-9f42-60f76c8c5975-3_1070_754_255_699} The diagram shows a solid cylinder standing on a horizontal circular base, centre \(O\) and radius 4 units. The line \(B A\) is a diameter and the radius \(O C\) is at \(90 ^ { \circ }\) to \(O A\). Points \(O ^ { \prime } , A ^ { \prime } , B ^ { \prime }\) and \(C ^ { \prime }\) lie on the upper surface of the cylinder such that \(O O ^ { \prime } , A A ^ { \prime } , B B ^ { \prime }\) and \(C C ^ { \prime }\) are all vertical and of length 12 units. The mid-point of \(B B ^ { \prime }\) is \(M\). Unit vectors \(\mathbf { i } , \mathbf { j }\) and \(\mathbf { k }\) are parallel to \(O A , O C\) and \(O O ^ { \prime }\) respectively.

1. Express each of the vectors \(\overrightarrow { M O }\) and \(\overrightarrow { M C ^ { \prime } }\) in terms of \(\mathbf { i } , \mathbf { j }\) and \(\mathbf { k }\).
2. Hence find the angle \(O M C ^ { \prime }\).

8 \includegraphics[max width=\textwidth, alt={}, center]{cbcb15b4-1870-4dfd-b6e9-839aa4601511-3_517_1117_1362_514} The diagram shows the roof of a house. The base of the roof, \(O A B C\), is rectangular and horizontal with \(O A = C B = 14 \mathrm {~m}\) and \(O C = A B = 8 \mathrm {~m}\). The top of the roof \(D E\) is 5 m above the base and \(D E = 6 \mathrm {~m}\). The sloping edges \(O D , C D , A E\) and \(B E\) are all equal in length. Unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are parallel to \(O A\) and \(O C\) respectively and the unit vector \(\mathbf { k }\) is vertically upwards.

1. Express the vector \(\overrightarrow { O D }\) in terms of \(\mathbf { i } , \mathbf { j }\) and \(\mathbf { k }\), and find its magnitude.
2. Use a scalar product to find angle \(D O B\).

4 \includegraphics[max width=\textwidth, alt={}, center]{53839c8c-07ea-4545-9c00-a6884aa2afc3-2_750_855_902_646} The diagram shows a prism \(A B C D P Q R S\) with a horizontal square base \(A P S D\) with sides of length 6 cm . The cross-section \(A B C D\) is a trapezium and is such that the vertical edges \(A B\) and \(D C\) are of lengths 5 cm and 2 cm respectively. Unit vectors \(\mathbf { i } , \mathbf { j }\) and \(\mathbf { k }\) are parallel to \(A D , A P\) and \(A B\) respectively.

1. Express each of the vectors \(\overrightarrow { C P }\) and \(\overrightarrow { C Q }\) in terms of \(\mathbf { i } , \mathbf { j }\) and \(\mathbf { k }\).
2. Use a scalar product to calculate angle \(P C Q\).

5 \includegraphics[max width=\textwidth, alt={}, center]{d68c82ec-8c85-40b9-8e81-bd53c7f8dafe-2_748_1155_1146_495} In the diagram, \(O A B C D E F G\) is a rectangular block in which \(O A = O D = 6 \mathrm {~cm}\) and \(A B = 12 \mathrm {~cm}\). The unit vectors \(\mathbf { i } , \mathbf { j }\) and \(\mathbf { k }\) are parallel to \(\overrightarrow { O A } , \overrightarrow { O C }\) and \(\overrightarrow { O D }\) respectively. The point \(P\) is the mid-point of \(D G , Q\) is the centre of the square face \(C B F G\) and \(R\) lies on \(A B\) such that \(A R = 4 \mathrm {~cm}\).

1. Express each of the vectors \(\overrightarrow { P Q }\) and \(\overrightarrow { R Q }\) in terms of \(\mathbf { i } , \mathbf { j }\) and \(\mathbf { k }\).
2. Use a scalar product to find angle \(R Q P\).

9 \includegraphics[max width=\textwidth, alt={}, center]{ea402a1d-3632-4637-9198-2365715b5246-14_670_857_260_644} The diagram shows a pyramid \(O A B C D\) with a horizontal rectangular base \(O A B C\). The sides \(O A\) and \(A B\) have lengths of 8 units and 6 units respectively. The point \(E\) on \(O B\) is such that \(O E = 2\) units. The point \(D\) of the pyramid is 7 units vertically above \(E\). Unit vectors \(\mathbf { i } , \mathbf { j }\) and \(\mathbf { k }\) are parallel to \(O A\), \(O C\) and \(E D\) respectively.

1. Show that \(\overrightarrow { O E } = 1.6 \mathbf { i } + 1.2 \mathbf { j }\).
2. Use a scalar product to find angle \(B D O\).

7 \includegraphics[max width=\textwidth, alt={}, center]{ebf16cae-1e80-44d2-9c51-630f5dc3c11f-12_775_823_260_662} The diagram shows a three-dimensional shape in which the base \(O A B C\) and the upper surface \(D E F G\) are identical horizontal squares. The parallelograms \(O A E D\) and \(C B F G\) both lie in vertical planes. The point \(M\) is the mid-point of \(A F\). Unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are parallel to \(O A\) and \(O C\) respectively and the unit vector \(\mathbf { k }\) is vertically upwards. The position vectors of \(A\) and \(D\) are given by \(\overrightarrow { O A } = 8 \mathbf { i }\) and \(\overrightarrow { O D } = 3 \mathbf { i } + 10 \mathbf { k }\).

1. Express each of the vectors \(\overrightarrow { A M }\) and \(\overrightarrow { G M }\) in terms of \(\mathbf { i } , \mathbf { j }\) and \(\mathbf { k }\).
2. Use a scalar product to find angle \(G M A\) correct to the nearest degree.

6 \includegraphics[max width=\textwidth, alt={}, center]{f462c036-45d3-4679-ad53-4edbf99df76d-08_739_867_260_641} The diagram shows a solid figure \(A B C D E F\) in which the horizontal base \(A B C\) is a triangle right-angled at \(A\). The lengths of \(A B\) and \(A C\) are 8 units and 4 units respectively and \(M\) is the mid-point of \(A B\). The point \(D\) is 7 units vertically above \(A\). Triangle \(D E F\) lies in a horizontal plane with \(D E , D F\) and \(F E\) parallel to \(A B , A C\) and \(C B\) respectively and \(N\) is the mid-point of \(F E\). The lengths of \(D E\) and \(D F\) are 4 units and 2 units respectively. Unit vectors \(\mathbf { i } , \mathbf { j }\) and \(\mathbf { k }\) are parallel to \(\overrightarrow { A B } , \overrightarrow { A C }\) and \(\overrightarrow { A D }\) respectively.

1. Find \(\overrightarrow { M F }\) in terms of \(\mathbf { i } , \mathbf { j }\) and \(\mathbf { k }\).
2. Find \(\overrightarrow { F N }\) in terms of \(\mathbf { i }\) and \(\mathbf { j }\).
3. Find \(\overrightarrow { M N }\) in terms of \(\mathbf { i } , \mathbf { j }\) and \(\mathbf { k }\).
4. Use a scalar product to find angle \(F M N\).

7 \includegraphics[max width=\textwidth, alt={}, center]{0f58de6c-aba7-4a79-a962-c23be3ee0aa9-3_529_698_260_721} The diagram shows a pyramid \(O A B C\) with a horizontal triangular base \(O A B\) and vertical height \(O C\). Angles \(A O B , B O C\) and \(A O C\) are each right angles. Unit vectors \(\mathbf { i } , \mathbf { j }\) and \(\mathbf { k }\) are parallel to \(O A , O B\) and \(O C\) respectively, with \(O A = 4\) units, \(O B = 2.4\) units and \(O C = 3\) units. The point \(P\) on \(C A\) is such that \(C P = 3\) units.

1. Show that \(\overrightarrow { C P } = 2.4 \mathbf { i } - 1.8 \mathbf { k }\).
2. Express \(\overrightarrow { O P }\) and \(\overrightarrow { B P }\) in terms of \(\mathbf { i } , \mathbf { j }\) and \(\mathbf { k }\).
3. Use a scalar product to find angle \(B P C\).

10 \includegraphics[max width=\textwidth, alt={}, center]{e753f588-97bc-4c6a-a82b-7b6a6d0cadc4-4_597_693_274_726} The diagram shows a cube \(O A B C D E F G\) in which the length of each side is 4 units. The unit vectors \(\mathbf { i } , \mathbf { j }\) and \(\mathbf { k }\) are parallel to \(\overrightarrow { O A } , \overrightarrow { O C }\) and \(\overrightarrow { O D }\) respectively. The mid-points of \(O A\) and \(D G\) are \(P\) and \(Q\) respectively and \(R\) is the centre of the square face \(A B F E\).

1. Express each of the vectors \(\overrightarrow { P R }\) and \(\overrightarrow { P Q }\) in terms of \(\mathbf { i } , \mathbf { j }\) and \(\mathbf { k }\).
2. Use a scalar product to find angle \(Q P R\).
3. Find the perimeter of triangle \(P Q R\), giving your answer correct to 1 decimal place.

4 \includegraphics[max width=\textwidth, alt={}, center]{08729aab-586b-4210-94c9-77b1f6b1d873-2_558_1488_863_331} The diagram shows a semicircular prism with a horizontal rectangular base \(A B C D\). The vertical ends \(A E D\) and \(B F C\) are semicircles of radius 6 cm . The length of the prism is 20 cm . The mid-point of \(A D\) is the origin \(O\), the mid-point of \(B C\) is \(M\) and the mid-point of \(D C\) is \(N\). The points \(E\) and \(F\) are the highest points of the semicircular ends of the prism. The point \(P\) lies on \(E F\) such that \(E P = 8 \mathrm {~cm}\). Unit vectors \(\mathbf { i } , \mathbf { j }\) and \(\mathbf { k }\) are parallel to \(O D , O M\) and \(O E\) respectively.

1. Express each of the vectors \(\overrightarrow { P A }\) and \(\overrightarrow { P N }\) in terms of \(\mathbf { i } , \mathbf { j }\) and \(\mathbf { k }\).
2. Use a scalar product to calculate angle \(A P N\).

6 \includegraphics[max width=\textwidth, alt={}, center]{b566719c-216e-41e5-8431-da77e1dad73e-2_590_666_1720_737} In the diagram, \(O A B C D E F G\) is a cube in which each side has length 6 . Unit vectors \(\mathbf { i } , \mathbf { j }\) and \(\mathbf { k }\) are parallel to \(\overrightarrow { O A } , \overrightarrow { O C }\) and \(\overrightarrow { O D }\) respectively. The point \(P\) is such that \(\overrightarrow { A P } = \frac { 1 } { 3 } \overrightarrow { A B }\) and the point \(Q\) is the mid-point of \(D F\).

1. Express each of the vectors \(\overrightarrow { O Q }\) and \(\overrightarrow { P Q }\) in terms of \(\mathbf { i } , \mathbf { j }\) and \(\mathbf { k }\).
2. Find the angle \(O Q P\).

9 \includegraphics[max width=\textwidth, alt={}, center]{ae57d8f1-5a0d-426c-952d-e8b99c6aeaba-4_582_1072_255_541} The diagram shows a pyramid \(O A B C P\) in which the horizontal base \(O A B C\) is a square of side 10 cm and the vertex \(P\) is 10 cm vertically above \(O\). The points \(D , E , F , G\) lie on \(O P , A P , B P , C P\) respectively and \(D E F G\) is a horizontal square of side 6 cm . The height of \(D E F G\) above the base is \(a \mathrm {~cm}\). Unit vectors \(\mathbf { i } , \mathbf { j }\) and \(\mathbf { k }\) are parallel to \(O A , O C\) and \(O D\) respectively.

1. Show that \(a = 4\).
2. Express the vector \(\overrightarrow { B G }\) in terms of \(\mathbf { i } , \mathbf { j }\) and \(\mathbf { k }\).
3. Use a scalar product to find angle \(G B A\).

3 \includegraphics[max width=\textwidth, alt={}, center]{02da6b6a-6db1-4bc3-ad4e-537e4f61dcac-2_397_949_657_596} The diagram shows a pyramid \(O A B C D\) in which the vertical edge \(O D\) is 3 units in length. The point \(E\) is the centre of the horizontal rectangular base \(O A B C\). The sides \(O A\) and \(A B\) have lengths of 6 units and 4 units respectively. The unit vectors \(\mathbf { i } , \mathbf { j }\) and \(\mathbf { k }\) are parallel to \(\overrightarrow { O A } , \overrightarrow { O C }\) and \(\overrightarrow { O D }\) respectively.

1. Express each of the vectors \(\overrightarrow { D B }\) and \(\overrightarrow { D E }\) in terms of \(\mathbf { i } , \mathbf { j }\) and \(\mathbf { k }\).
2. Use a scalar product to find angle \(B D E\).

4 \includegraphics[max width=\textwidth, alt={}, center]{16a5835e-002f-4c49-aacf-cda41c37f214-2_711_643_900_753} The diagram shows a pyramid \(O A B C\) in which the edge \(O C\) is vertical. The horizontal base \(O A B\) is a triangle, right-angled at \(O\), and \(D\) is the mid-point of \(A B\). The edges \(O A , O B\) and \(O C\) have lengths of 8 units, 6 units and 10 units respectively. The unit vectors \(\mathbf { i } , \mathbf { j }\) and \(\mathbf { k }\) are parallel to \(\overrightarrow { O A } , \overrightarrow { O B }\) and \(\overrightarrow { O C }\) respectively.

1. Express each of the vectors \(\overrightarrow { O D }\) and \(\overrightarrow { C D }\) in terms of \(\mathbf { i } , \mathbf { j }\) and \(\mathbf { k }\).
2. Use a scalar product to find angle ODC.

8 \includegraphics[max width=\textwidth, alt={}, center]{d178603a-f59a-4986-b5ab-b47eceedb2fc-12_595_748_260_699} The diagram shows a solid figure \(O A B C D E F\) having a horizontal rectangular base \(O A B C\) with \(O A = 6\) units and \(A B = 3\) units. The vertical edges \(O F , A D\) and \(B E\) have lengths 6 units, 4 units and 4 units respectively. Unit vectors \(\mathbf { i } , \mathbf { j }\) and \(\mathbf { k }\) are parallel to \(O A , O C\) and \(O F\) respectively.

1. Find \(\overrightarrow { D F }\).
2. Find the unit vector in the direction of \(\overrightarrow { E F }\).
3. Use a scalar product to find angle \(E F D\).

7 \includegraphics[max width=\textwidth, alt={}, center]{2d5f452d-f820-40fc-9e22-9d3ac4f0698b-10_819_497_262_826} The diagram shows a solid cylinder standing on a horizontal circular base with centre \(O\) and radius 4 units. Points \(A , B\) and \(C\) lie on the circumference of the base such that \(A B\) is a diameter and angle \(B O C = 90 ^ { \circ }\). Points \(P , Q\) and \(R\) lie on the upper surface of the cylinder vertically above \(A , B\) and \(C\) respectively. The height of the cylinder is 12 units. The mid-point of \(C R\) is \(M\) and \(N\) lies on \(B Q\) with \(B N = 4\) units. Unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are parallel to \(O B\) and \(O C\) respectively and the unit vector \(\mathbf { k }\) is vertically upwards.
Evaluate \(\overrightarrow { P N } \cdot \overrightarrow { P M }\) and hence find angle \(M P N\). \includegraphics[max width=\textwidth, alt={}, center]{2d5f452d-f820-40fc-9e22-9d3ac4f0698b-12_483_574_262_788} The diagram shows an isosceles triangle \(A C B\) in which \(A B = B C = 8 \mathrm {~cm}\) and \(A C = 12 \mathrm {~cm}\). The arc \(X C\) is part of a circle with centre \(A\) and radius 12 cm , and the arc \(Y C\) is part of circle with centre \(B\) and radius 8 cm . The points \(A , B , X\) and \(Y\) lie on a straight line.

1. Show that angle \(C B Y = 1.445\) radians, correct to 4 significant figures.
2. Find the perimeter of the shaded region.

6 \includegraphics[max width=\textwidth, alt={}, center]{d5d94eb8-7f41-4dff-b503-8be4f20e21b7-08_743_897_260_623} The diagram shows a solid figure \(O A B C D E F G\) with a horizontal rectangular base \(O A B C\) in which \(O A = 8\) units and \(A B = 6\) units. The rectangle \(D E F G\) lies in a horizontal plane and is such that \(D\) is 7 units vertically above \(O\) and \(D E\) is parallel to \(O A\). The sides \(D E\) and \(D G\) have lengths 4 units and 2 units respectively. Unit vectors \(\mathbf { i } , \mathbf { j }\) and \(\mathbf { k }\) are parallel to \(O A , O C\) and \(O D\) respectively. Use a scalar product to find angle \(O B F\), giving your answer in the form \(\cos ^ { - 1 } \left( \frac { a } { b } \right)\), where \(a\) and \(b\) are integers.

7 \includegraphics[max width=\textwidth, alt={}, center]{567c3d72-c633-4ae0-8605-f63f93d718c4-12_784_677_260_735} The diagram shows a three-dimensional shape \(O A B C D E F G\). The base \(O A B C\) and the upper surface \(D E F G\) are identical horizontal rectangles. The parallelograms \(O A E D\) and \(C B F G\) both lie in vertical planes. Points \(P\) and \(Q\) are the mid-points of \(O D\) and \(G F\) respectively. Unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are parallel to \(\overrightarrow { O A }\) and \(\overrightarrow { O C }\) respectively and the unit vector \(\mathbf { k }\) is vertically upwards. The position vectors of \(A , C\) and \(D\) are given by \(\overrightarrow { O A } = 6 \mathbf { i } , \overrightarrow { O C } = 8 \mathbf { j }\) and \(\overrightarrow { O D } = 2 \mathbf { i } + 10 \mathbf { k }\).

1. Express each of the vectors \(\overrightarrow { P B }\) and \(\overrightarrow { P Q }\) in terms of \(\mathbf { i } , \mathbf { j }\) and \(\mathbf { k }\).
2. Determine whether \(P\) is nearer to \(Q\) or to \(B\).
3. Use a scalar product to find angle \(B P Q\).

9 \includegraphics[max width=\textwidth, alt={}, center]{6eada775-be31-4d0f-87b5-0e7e00b376f3-14_670_857_260_644} The diagram shows a pyramid \(O A B C D\) with a horizontal rectangular base \(O A B C\). The sides \(O A\) and \(A B\) have lengths of 8 units and 6 units respectively. The point \(E\) on \(O B\) is such that \(O E = 2\) units. The point \(D\) of the pyramid is 7 units vertically above \(E\). Unit vectors \(\mathbf { i } , \mathbf { j }\) and \(\mathbf { k }\) are parallel to \(O A\), \(O C\) and \(E D\) respectively.

1. Show that \(\overrightarrow { O E } = 1.6 \mathbf { i } + 1.2 \mathbf { j }\).
2. Use a scalar product to find angle \(B D O\).

7 A glass ornament OABCDEFG is a truncated pyramid on a rectangular base (see Fig. 7). All dimensions are in centimetres. \begin{figure}[h] \end{figure}

1. Write down the vectors \(\overrightarrow { \mathrm { CD } }\) and \(\overrightarrow { \mathrm { CB } }\).
2. Find the length of the edge CD.
3. Show that the vector \(4 \mathbf { i } + \mathbf { k }\) is perpendicular to the vectors \(\overrightarrow { \mathrm { CD } }\) and \(\overrightarrow { \mathrm { CB } }\). Hence find the cartesian equation of the plane BCDE .
4. Write down vector equations for the lines OG and AF . Show that they meet at the point P with coordinates (5, 10, 40). You may assume that the lines CD and BE also meet at the point P .
   The volume of a pyramid is \(\frac { 1 } { 3 } \times\) area of base × height.
5. Find the volumes of the pyramids POABC and PDEFG . Hence find the volume of the ornament.

1 A glass ornament OABCDEFG is a truncated pyramid on a rectangular base (see Fig. 7). All dimensions are in centimetres. \begin{figure}[h] \end{figure}

1. Write down the vectors \(\overrightarrow { \mathrm { CD } }\) and \(\overrightarrow { \mathrm { CB } }\).
2. Find the length of the edge CD.
3. Show that the vector \(4 \mathbf { i } + \mathbf { k }\) is perpendicular to the vectors \(\overrightarrow { \mathrm { CD } }\) and \(\overrightarrow { \mathrm { CB } }\). Hence find the cartesian equation of the plane BCDE.
4. Write down vector equations for the lines OG and AF . Show that they meet at the point P with coordinates (5, 10, 40). You may assume that the lines CD and BE also meet at the point P .
   The volume of a pyramid is \(\frac { 1 } { 3 } \times\) area of base × height.
5. Find the volumes of the pyramids POABC and PDEFG . Hence find the volume of the ornament.