1.10b Vectors in 3D: i,j,k notation

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 The points \(A , B , C\) and \(D\) have position vectors \(3 \mathbf { i } + 2 \mathbf { k } , 2 \mathbf { i } - 2 \mathbf { j } + 5 \mathbf { k } , 2 \mathbf { j } + 7 \mathbf { k }\) and \(- 2 \mathbf { i } + 10 \mathbf { j } + 7 \mathbf { k }\) respectively.

1. Use a scalar product to show that \(B A\) and \(B C\) are perpendicular.
2. Show that \(B C\) and \(A D\) are parallel and find the ratio of the length of \(B C\) to the length of \(A D\).

9 Relative to an origin \(O\), the position vectors of the points \(A , B , C\) and \(D\) are given by $$\overrightarrow { O A } = \left( \begin{array} { r } 1 \\ 3 \\ - 1 \end{array} \right) , \quad \overrightarrow { O B } = \left( \begin{array} { r } 3 \\ - 1 \\ 3 \end{array} \right) , \quad \overrightarrow { O C } = \left( \begin{array} { l } 4 \\ 2 \\ p \end{array} \right) \quad \text { and } \quad \overrightarrow { O D } = \left( \begin{array} { r } - 1 \\ 0 \\ q \end{array} \right) ,$$ where \(p\) and \(q\) are constants. Find

1. the unit vector in the direction of \(\overrightarrow { A B }\),
2. the value of \(p\) for which angle \(A O C = 90 ^ { \circ }\),
3. the values of \(q\) for which the length of \(\overrightarrow { A D }\) is 7 units.

11 Relative to an origin \(O\), the position vectors of the points \(A\) and \(B\) are given by $$\overrightarrow { O A } = 2 \mathbf { i } + 3 \mathbf { j } - \mathbf { k } \quad \text { and } \quad \overrightarrow { O B } = 4 \mathbf { i } - 3 \mathbf { j } + 2 \mathbf { k }$$

1. Use a scalar product to find angle \(A O B\), correct to the nearest degree.
2. Find the unit vector in the direction of \(\overrightarrow { A B }\).
3. The point \(C\) is such that \(\overrightarrow { O C } = 6 \mathbf { j } + p \mathbf { k }\), where \(p\) is a constant. Given that the lengths of \(\overrightarrow { A B }\) and \(\overrightarrow { A C }\) are equal, find the possible values of \(p\).

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 \(R Q P\).

7 \includegraphics[max width=\textwidth, alt={}, center]{1a4ddaa9-1ec2-4138-bfcb-a482fe6c942f-3_394_750_260_699} The diagram shows a trapezium \(A B C D\) in which \(B A\) is parallel to \(C D\). The position vectors of \(A , B\) and \(C\) relative to an origin \(O\) are given by $$\overrightarrow { O A } = \left( \begin{array} { l } 3 \\ 4 \\ 0 \end{array} \right) , \quad \overrightarrow { O B } = \left( \begin{array} { l } 1 \\ 3 \\ 2 \end{array} \right) \quad \text { and } \quad \overrightarrow { O C } = \left( \begin{array} { l } 4 \\ 5 \\ 6 \end{array} \right)$$

1. Use a scalar product to show that \(A B\) is perpendicular to \(B C\).
2. Given that the length of \(C D\) is 12 units, find the position vector of \(D\).

7 The position vectors of points \(A , B\) and \(C\) relative to an origin \(O\) are given by $$\overrightarrow { O A } = \left( \begin{array} { l } 2 \\ 1 \\ 3 \end{array} \right) , \quad \overrightarrow { O B } = \left( \begin{array} { r } 6 \\ - 1 \\ 7 \end{array} \right) \quad \text { and } \quad \overrightarrow { O C } = \left( \begin{array} { l } 2 \\ 4 \\ 7 \end{array} \right)$$

1. Show that angle \(B A C = \cos ^ { - 1 } \left( \frac { 1 } { 3 } \right)\).
2. Use the result in part (i) to find the exact value of the area of triangle \(A B C\).

9 The position vectors of \(A , B\) and \(C\) relative to an origin \(O\) are given by $$\overrightarrow { O A } = \left( \begin{array} { r } 2 \\ 3 \\ - 4 \end{array} \right) , \quad \overrightarrow { O B } = \left( \begin{array} { c } 1 \\ 5 \\ p \end{array} \right) \quad \text { and } \quad \overrightarrow { O C } = \left( \begin{array} { l } 5 \\ 0 \\ 2 \end{array} \right) ,$$ where \(p\) is a constant.

1. Find the value of \(p\) for which the lengths of \(A B\) and \(C B\) are equal.
2. For the case where \(p = 1\), use a scalar product to find angle \(A B C\).

2 Relative to an origin \(O\), the position vectors of points \(A\) and \(B\) are given by $$\overrightarrow { O A } = \left( \begin{array} { r } 3 \\ - 6 \\ p \end{array} \right) \quad \text { and } \quad \overrightarrow { O B } = \left( \begin{array} { r } 2 \\ - 6 \\ - 7 \end{array} \right)$$ and angle \(A O B = 90 ^ { \circ }\).

1. Find the value of \(p\). \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) The point \(C\) is such that \(\overrightarrow { O C } = \frac { 2 } { 3 } \overrightarrow { O A }\).
2. Find the unit vector in the direction of \(\overrightarrow { B C }\). \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\)

8 Relative to an origin \(O\), the position vectors of three points \(A , B\) and \(C\) are given by $$\overrightarrow { O A } = 3 \mathbf { i } + p \mathbf { j } - 2 p \mathbf { k } , \quad \overrightarrow { O B } = 6 \mathbf { i } + ( p + 4 ) \mathbf { j } + 3 \mathbf { k } \quad \text { and } \quad \overrightarrow { O C } = ( p - 1 ) \mathbf { i } + 2 \mathbf { j } + q \mathbf { k }$$ where \(p\) and \(q\) are constants.

1. In the case where \(p = 2\), use a scalar product to find angle \(A O B\).
2. In the case where \(\overrightarrow { A B }\) is parallel to \(\overrightarrow { O C }\), find the values of \(p\) and \(q\).

7 Relative to an origin \(O\), the position vectors of the points \(A , B\) and \(C\) are given by $$\overrightarrow { O A } = \left( \begin{array} { r } 1 \\ - 3 \\ 2 \end{array} \right) , \quad \overrightarrow { O B } = \left( \begin{array} { r } - 1 \\ 3 \\ 5 \end{array} \right) \quad \text { and } \quad \overrightarrow { O C } = \left( \begin{array} { r } 3 \\ 1 \\ - 2 \end{array} \right)$$

1. Find \(\overrightarrow { A C }\).
2. The point \(M\) is the mid-point of \(A C\). Find the unit vector in the direction of \(\overrightarrow { O M }\).
3. Evaluate \(\overrightarrow { A B } \cdot \overrightarrow { A C }\) and hence find angle \(B A C\).

4 The function f is such that \(\mathrm { f } ( x ) = a + b \cos x\) for \(0 \leqslant x \leqslant 2 \pi\). It is given that \(\mathrm { f } \left( \frac { 1 } { 3 } \pi \right) = 5\) and \(\mathrm { f } ( \pi ) = 11\).

1. Find the values of the constants \(a\) and \(b\). \includegraphics[max width=\textwidth, alt={}, center]{58d65166-2b1a-4b58-9859-afe919c0a3a9-05_63_1566_397_328}
2. Find the set of values of \(k\) for which the equation \(\mathrm { f } ( x ) = k\) has no solution. \includegraphics[max width=\textwidth, alt={}, center]{58d65166-2b1a-4b58-9859-afe919c0a3a9-06_622_878_260_632} The diagram shows a three-dimensional shape. The base \(O A B\) is a horizontal triangle in which angle \(A O B\) is \(90 ^ { \circ }\). The side \(O B C D\) is a rectangle and the side \(O A D\) lies in a vertical plane. Unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are parallel to \(O A\) and \(O B\) respectively and the unit vector \(\mathbf { k }\) is vertical. The position vectors of \(A , B\) and \(D\) are given by \(\overrightarrow { O A } = 8 \mathbf { i } , \overrightarrow { O B } = 5 \mathbf { j }\) and \(\overrightarrow { O D } = 2 \mathbf { i } + 4 \mathbf { k }\).

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.

8 The position vectors of points \(A\) and \(B\), relative to an origin \(O\), are given by $$\overrightarrow { O A } = \left( \begin{array} { r } 6 \\ - 2 \\ - 6 \end{array} \right) \quad \text { and } \quad \overrightarrow { O B } = \left( \begin{array} { r } 3 \\ k \\ - 3 \end{array} \right)$$ where \(k\) is a constant.

1. Find the value of \(k\) for which angle \(A O B\) is \(90 ^ { \circ }\).
2. Find the values of \(k\) for which the lengths of \(O A\) and \(O B\) are equal.
   The point \(C\) is such that \(\overrightarrow { A C } = 2 \overrightarrow { C B }\).
3. In the case where \(k = 4\), find the unit vector in the direction of \(\overrightarrow { O C }\).

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 \(A P N\).

8 The first term of an arithmetic progression is 8 and the common difference is \(d\), where \(d \neq 0\). The first term, the fifth term and the eighth term of this arithmetic progression are the first term, the second term and the third term, respectively, of a geometric progression whose common ratio is \(r\).

1. Write down two equations connecting \(d\) and \(r\). Hence show that \(r = \frac { 3 } { 4 }\) and find the value of \(d\).
2. Find the sum to infinity of the geometric progression.
3. Find the sum of the first 8 terms of the arithmetic progression.

5

1. Prove the identity \(( \sin x + \cos x ) ( 1 - \sin x \cos x ) \equiv \sin ^ { 3 } x + \cos ^ { 3 } x\).
2. Solve the equation \(( \sin x + \cos x ) ( 1 - \sin x \cos x ) = 9 \sin ^ { 3 } x\) for \(0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }\).

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 \(G B A\).

10 \includegraphics[max width=\textwidth, alt={}, center]{32a57386-2696-4fda-a3cb-ca0c5c3be432-4_561_599_744_774} The diagram shows triangle \(O A B\), in which the position vectors of \(A\) and \(B\) with respect to \(O\) are given by $$\overrightarrow { O A } = 2 \mathbf { i } + \mathbf { j } - 3 \mathbf { k } \quad \text { and } \quad \overrightarrow { O B } = - 3 \mathbf { i } + 2 \mathbf { j } - 4 \mathbf { k } .$$ \(C\) is a point on \(O A\) such that \(\overrightarrow { O C } = p \overrightarrow { O A }\), where \(p\) is a constant.

1. Find angle \(A O B\).
2. Find \(\overrightarrow { B C }\) in terms of \(p\) and vectors \(\mathbf { i } , \mathbf { j }\) and \(\mathbf { k }\).
3. Find the value of \(p\) given that \(B C\) is perpendicular to \(O A\).

8 Relative to an origin \(O\), the point \(A\) has position vector \(4 \mathbf { i } + 7 \mathbf { j } - p \mathbf { k }\) and the point \(B\) has position vector \(8 \mathbf { i } - \mathbf { j } - p \mathbf { k }\), where \(p\) is a constant.

1. Find \(\overrightarrow { O A } \cdot \overrightarrow { O B }\).
2. Hence show that there are no real values of \(p\) for which \(O A\) and \(O B\) are perpendicular to each other.
3. Find the values of \(p\) for which angle \(A O B = 60 ^ { \circ }\).