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

3 Relative to an origin \(O\), the position vectors of points \(A\) and \(B\) are given by $$\overrightarrow { O A } = 5 \mathbf { i } + \mathbf { j } + 2 \mathbf { k } \quad \text { and } \quad \overrightarrow { O B } = 2 \mathbf { i } + 7 \mathbf { j } + p \mathbf { k }$$ where \(p\) is a constant.

1. Find the value of \(p\) for which angle \(A O B\) is \(90 ^ { \circ }\).
2. In the case where \(p = 4\), find the vector which has magnitude 28 and is in the same direction as \(\overrightarrow { A B }\).

9 The position vectors of points \(A\) and \(B\) relative to an origin \(O\) are \(\mathbf { a }\) and \(\mathbf { b }\) respectively. The position vectors of points \(C\) and \(D\) relative to \(O\) are \(3 \mathbf { a }\) and \(2 \mathbf { b }\) respectively. It is given that $$\mathbf { a } = \left( \begin{array} { l } 2 \\ 1 \\ 2 \end{array} \right) \quad \text { and } \quad \mathbf { b } = \left( \begin{array} { l } 4 \\ 0 \\ 6 \end{array} \right) .$$

1. Find the unit vector in the direction of \(\overrightarrow { C D }\).
2. The point \(E\) is the mid-point of \(C D\). Find angle \(E O D\).

7 The position vectors of the points \(A\) and \(B\), relative to an origin \(O\), are given by $$\overrightarrow { O A } = \left( \begin{array} { l } 1 \\ 0 \\ 2 \end{array} \right) \quad \text { and } \quad \overrightarrow { O B } = \left( \begin{array} { r } k \\ - k \\ 2 k \end{array} \right)$$ where \(k\) is a constant.

1. In the case where \(k = 2\), calculate angle \(A O B\).
2. Find the values of \(k\) for which \(\overrightarrow { A B }\) is a unit vector.

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

1. In the case where \(O A B\) is a straight line, state the value of \(p\) and find the unit vector in the direction of \(\overrightarrow { O A }\).
2. In the case where \(O A\) is perpendicular to \(A B\), find the possible values of \(p\).
3. In the case where \(p = 3\), the point \(C\) is such that \(O A B C\) is a parallelogram. Find the position vector of \(C\).

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

1. In the case where \(p = 6\), find the unit vector in the direction of \(\overrightarrow { A B }\).
2. Find the values of \(p\) for which angle \(A O B = \cos ^ { - 1 } \left( \frac { 1 } { 5 } \right)\).

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.

6 Relative to an origin \(O\), the position vector of \(A\) is \(3 \mathbf { i } + 2 \mathbf { j } - \mathbf { k }\) and the position vector of \(B\) is \(7 \mathbf { i } - 3 \mathbf { j } + \mathbf { k }\).

1. Show that angle \(O A B\) is a right angle.
2. Find the area of triangle \(O A B\).

7 Relative to an origin \(O\), the position vectors of points \(A , B\) and \(C\) are given by $$\overrightarrow { O A } = \left( \begin{array} { r } 0 \\ 2 \\ - 3 \end{array} \right) , \quad \overrightarrow { O B } = \left( \begin{array} { r } 2 \\ 5 \\ - 2 \end{array} \right) \quad \text { and } \quad \overrightarrow { O C } = \left( \begin{array} { l } 3 \\ p \\ q \end{array} \right)$$

1. In the case where \(A B C\) is a straight line, find the values of \(p\) and \(q\).
2. In the case where angle \(B A C\) is \(90 ^ { \circ }\), express \(q\) in terms of \(p\).
3. In the case where \(p = 3\) and the lengths of \(A B\) and \(A C\) are equal, find the possible values of \(q\).

5 Relative to an origin \(O\), the position vectors of the points \(A\) and \(B\) are given by $$\overrightarrow { O A } = \left( \begin{array} { c } p - 6 \\ 2 p - 6 \\ 1 \end{array} \right) \quad \text { and } \quad \overrightarrow { O B } = \left( \begin{array} { c } 4 - 2 p \\ p \\ 2 \end{array} \right)$$ where \(p\) is a constant.

1. For the case where \(O A\) is perpendicular to \(O B\), find the value of \(p\).
2. For the case where \(O A B\) is a straight line, find the vectors \(\overrightarrow { O A }\) and \(\overrightarrow { O B }\). Find also the length of the line \(O A\).

9 \includegraphics[max width=\textwidth, alt={}, center]{9f17f7b8-b54d-467d-be26-21c599ce6ca2-4_724_1488_257_330} The diagram shows a cuboid \(O A B C D E F G\) with a horizontal base \(O A B C\) in which \(O A = 4 \mathrm {~cm}\) and \(A B = 15 \mathrm {~cm}\). The height \(O D\) of the cuboid is 2 cm . The point \(X\) on \(A B\) is such that \(A X = 5 \mathrm {~cm}\) and the point \(P\) on \(D G\) is such that \(D P = p \mathrm {~cm}\), where \(p\) is a constant. Unit vectors \(\mathbf { i } , \mathbf { j }\) and \(\mathbf { k }\) are parallel to \(O A , O C\) and \(O D\) respectively.

1. Find the possible values of \(p\) such that angle \(O P X = 90 ^ { \circ }\).
2. For the case where \(p = 9\), find the unit vector in the direction of \(\overrightarrow { X P }\).
3. A point \(Q\) lies on the face \(C B F G\) and is such that \(X Q\) is parallel to \(A G\). Find \(\overrightarrow { X Q }\).

9 \includegraphics[max width=\textwidth, alt={}, center]{518bb805-5b14-4b41-94fd-38a31a90c218-16_533_601_258_772} The diagram shows a trapezium \(O A B C\) in which \(O A\) is parallel to \(C B\). The position vectors of \(A\) and \(B\) relative to the origin \(O\) are given by \(\overrightarrow { O A } = \left( \begin{array} { r } 2 \\ - 2 \\ - 1 \end{array} \right)\) and \(\overrightarrow { O B } = \left( \begin{array} { l } 6 \\ 1 \\ 1 \end{array} \right)\).

1. Show that angle \(O A B\) is \(90 ^ { \circ }\).
   The magnitude of \(\overrightarrow { C B }\) is three times the magnitude of \(\overrightarrow { O A }\).
2. Find the position vector of \(C\).
3. Find the exact area of the trapezium \(O A B C\), giving your answer in the form \(a \sqrt { } b\), where \(a\) and \(b\) are integers.

9 Relative to an origin \(O\), the position vectors of the points \(A , B\) and \(C\) are given by $$\overrightarrow { O A } = \left( \begin{array} { r } 8 \\ - 6 \\ 5 \end{array} \right) , \quad \overrightarrow { O B } = \left( \begin{array} { r } - 10 \\ 3 \\ - 13 \end{array} \right) \quad \text { and } \quad \overrightarrow { O C } = \left( \begin{array} { r } 2 \\ - 3 \\ - 1 \end{array} \right)$$ A fourth point, \(D\), is such that the magnitudes \(| \overrightarrow { A B } | , | \overrightarrow { B C } |\) and \(| \overrightarrow { C D } |\) are the first, second and third terms respectively of a geometric progression.

1. Find the magnitudes \(| \overrightarrow { A B } | , | \overrightarrow { B C } |\) and \(| \overrightarrow { C D } |\).
2. Given that \(D\) is a point lying on the line through \(B\) and \(C\), find the two possible position vectors of the point \(D\).

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 \(E F D\).

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.

10 \includegraphics[max width=\textwidth, alt={}, center]{0e4a249a-9e6a-49d4-996c-fe07b7730f59-16_318_1006_260_568} Relative to an origin \(O\), the position vectors of the points \(A , B , C\) and \(D\), shown in the diagram, are given by $$\overrightarrow { O A } = \left( \begin{array} { r } - 1 \\ 3 \\ - 4 \end{array} \right) , \quad \overrightarrow { O B } = \left( \begin{array} { r } 2 \\ - 3 \\ 5 \end{array} \right) , \quad \overrightarrow { O C } = \left( \begin{array} { r } 4 \\ - 2 \\ 5 \end{array} \right) \quad \text { and } \quad \overrightarrow { O D } = \left( \begin{array} { r } 2 \\ 2 \\ - 1 \end{array} \right) .$$

1. Show that \(A B\) is perpendicular to \(B C\).
2. Show that \(A B C D\) is a trapezium.
3. Find the area of \(A B C D\), giving your answer correct to 2 decimal places.

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 \(B P Q\).

10 Relative to an origin \(O\), the position vectors of the points \(A , B\) and \(X\) are given by $$\overrightarrow { O A } = \left( \begin{array} { r } - 8 \\ - 4 \\ 2 \end{array} \right) , \quad \overrightarrow { O B } = \left( \begin{array} { r } 10 \\ 2 \\ 11 \end{array} \right) \quad \text { and } \quad \overrightarrow { O X } = \left( \begin{array} { r } - 2 \\ - 2 \\ 5 \end{array} \right)$$

1. Find \(\overrightarrow { A X }\) and show that \(A X B\) is a straight line. \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) The position vector of a point \(C\) is given by \(\overrightarrow { O C } = \left( \begin{array} { r } 1 \\ - 8 \\ 3 \end{array} \right)\).
2. Show that \(C X\) is perpendicular to \(A X\). \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\)
3. Find the area of triangle \(A B C\). \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \includegraphics[max width=\textwidth, alt={}, center]{17e813c6-890f-4198-b20a-557b133e8c34-18_949_1087_260_529} The diagram shows part of the curve \(y = ( x - 1 ) ^ { - 2 } + 2\), and the lines \(x = 1\) and \(x = 3\). The point \(A\) on the curve has coordinates \(( 2,3 )\). The normal to the curve at \(A\) crosses the line \(x = 1\) at \(B\).
4. Show that the normal \(A B\) has equation \(y = \frac { 1 } { 2 } x + 2\). \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\)
5. Find, showing all necessary working, the volume of revolution obtained when the shaded region is rotated through \(360 ^ { \circ }\) about the \(x\)-axis. \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\)

8 The straight line \(l\) passes through the points \(A\) and \(B\) whose position vectors are \(\mathbf { i } + \mathbf { k }\) and \(4 \mathbf { i } - \mathbf { j } + 3 \mathbf { k }\) respectively. The plane \(p\) has equation \(x + 3 y - 2 z = 3\).

1. Given that \(l\) intersects \(p\), find the position vector of the point of intersection.
2. Find the equation of the plane which contains \(l\) and is perpendicular to \(p\), giving your answer in the form \(a x + b y + c z = 1\).

10 With respect to the origin \(O\), the points \(A\) and \(B\) have position vectors given by $$\overrightarrow { O A } = 2 \mathbf { i } + 2 \mathbf { j } + \mathbf { k } \quad \text { and } \quad \overrightarrow { O B } = \mathbf { i } + 4 \mathbf { j } + 3 \mathbf { k }$$ The line \(l\) has vector equation \(\mathbf { r } = 4 \mathbf { i } - 2 \mathbf { j } + 2 \mathbf { k } + s ( \mathbf { i } + 2 \mathbf { j } + \mathbf { k } )\).

1. Prove that the line \(I\) does not intersect the line through \(A\) and \(B\).
2. Find the equation of the plane containing \(l\) and the point \(A\), giving your answer in the form \(a x + b y + c z = d\).

9 \includegraphics[max width=\textwidth, alt={}, center]{8580dddb-cc72-4745-9e0f-1ac641c6506d-3_693_537_1206_804} The diagram shows a set of rectangular axes \(O x , O y\) and \(O z\), and three points \(A , B\) and \(C\) with position vectors \(\overrightarrow { O A } = \left( \begin{array} { l } 2 \\ 0 \\ 0 \end{array} \right) , \overrightarrow { O B } = \left( \begin{array} { l } 1 \\ 2 \\ 0 \end{array} \right)\) and \(\overrightarrow { O C } = \left( \begin{array} { l } 1 \\ 1 \\ 2 \end{array} \right)\).

1. Find the equation of the plane \(A B C\), giving your answer in the form \(a x + b y + c z = d\).
2. Calculate the acute angle between the planes \(A B C\) and \(O A B\).

10 The points \(A\) and \(B\) have position vectors, relative to the origin \(O\), given by $$\overrightarrow { O A } = \mathbf { i } + 2 \mathbf { j } + 3 \mathbf { k } \quad \text { and } \quad \overrightarrow { O B } = 2 \mathbf { i } + \mathbf { j } + 3 \mathbf { k } .$$ The line \(l\) has vector equation $$\mathbf { r } = ( 1 - 2 t ) \mathbf { i } + ( 5 + t ) \mathbf { j } + ( 2 - t ) \mathbf { k }$$

1. Show that \(l\) does not intersect the line passing through \(A\) and \(B\).
2. The point \(P\) lies on \(l\) and is such that angle \(P A B\) is equal to \(60 ^ { \circ }\). Given that the position vector of \(P\) is \(( 1 - 2 t ) \mathbf { i } + ( 5 + t ) \mathbf { j } + ( 2 - t ) \mathbf { k }\), show that \(3 t ^ { 2 } + 7 t + 2 = 0\). Hence find the only possible position vector of \(P\).

3 Points \(A\) and \(B\) have coordinates \(( - 1,2,5 )\) and \(( 2 , - 2,11 )\) respectively. The plane \(p\) passes through \(B\) and is perpendicular to \(A B\).

1. Find an equation of \(p\), giving your answer in the form \(a x + b y + c z = d\).
2. Find the acute angle between \(p\) and the \(y\)-axis.

6 The points \(P\) and \(Q\) have position vectors, relative to the origin \(O\), given by $$\overrightarrow { O P } = 7 \mathbf { i } + 7 \mathbf { j } - 5 \mathbf { k } \quad \text { and } \quad \overrightarrow { O Q } = - 5 \mathbf { i } + \mathbf { j } + \mathbf { k }$$ The mid-point of \(P Q\) is the point \(A\). The plane \(\Pi\) is perpendicular to the line \(P Q\) and passes through \(A\).

1. Find the equation of \(\Pi\), giving your answer in the form \(a x + b y + c z = d\).
2. The straight line through \(P\) parallel to the \(x\)-axis meets \(\Pi\) at the point \(B\). Find the distance \(A B\), correct to 3 significant figures.

10 The points \(A\) and \(B\) have position vectors given by \(\overrightarrow { O A } = \mathbf { i } - 2 \mathbf { j } + 2 \mathbf { k }\) and \(\overrightarrow { O B } = 3 \mathbf { i } + \mathbf { j } + \mathbf { k }\). The line \(l\) has equation \(\mathbf { r } = 2 \mathbf { i } + \mathbf { j } + m \mathbf { k } + \mu ( \mathbf { i } - 2 \mathbf { j } - 4 \mathbf { k } )\), where \(m\) is a constant.

1. Given that the line \(l\) intersects the line passing through \(A\) and \(B\), find the value of \(m\).
2. Find the equation of the plane which is parallel to \(\mathbf { i } - 2 \mathbf { j } - 4 \mathbf { k }\) and contains the points \(A\) and \(B\). Give your answer in the form \(a x + b y + c z = d\).