1.10e Position vectors: and displacement

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.

5 \includegraphics[max width=\textwidth, alt={}, center]{e439eea6-76f0-41eb-aa91-bd0f3e4e1a07-2_591_1061_1098_541} The diagram shows a rhombus \(A B C D\). The points \(B\) and \(D\) have coordinates \(( 2,10 )\) and \(( 6,2 )\) respectively, and \(A\) lies on the \(x\)-axis. The mid-point of \(B D\) is \(M\). Find, by calculation, the coordinates of each of \(M , A\) and \(C\).

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 \(p\).

10 Relative to an origin \(O\), the position vectors of points \(A\) and \(B\) are \(2 \mathbf { i } + \mathbf { j } + 2 \mathbf { k }\) and \(3 \mathbf { i } - 2 \mathbf { j } + p \mathbf { k }\) respectively.

1. Find the value of \(p\) for which \(O A\) and \(O B\) are perpendicular.
2. In the case where \(p = 6\), use a scalar product to find angle \(A O B\), correct to the nearest degree.
3. Express the vector \(\overrightarrow { A B }\) is terms of \(p\) and hence find the values of \(p\) for which the length of \(A B\) is 3.5 units.

6 Relative to an origin \(O\), the position vectors of the points \(A\) and \(B\) are given by $$\overrightarrow { O A } = 2 \mathbf { i } - 8 \mathbf { j } + 4 \mathbf { k } \quad \text { and } \quad \overrightarrow { O B } = 7 \mathbf { i } + 2 \mathbf { j } - \mathbf { k }$$

1. Find the value of \(\overrightarrow { O A } \cdot \overrightarrow { O B }\) and hence state whether angle \(A O B\) is acute, obtuse or a right angle.
2. The point \(X\) is such that \(\overrightarrow { A X } = \frac { 2 } { 5 } \overrightarrow { A B }\). Find the unit vector in the direction of \(O X\).

9 The coordinates of \(A\) are \(( - 3,2 )\) and the coordinates of \(C\) are (5,6). The mid-point of \(A C\) is \(M\) and the perpendicular bisector of \(A C\) cuts the \(x\)-axis at \(B\).

1. Find the equation of \(M B\) and the coordinates of \(B\).
2. Show that \(A B\) is perpendicular to \(B C\).
3. Given that \(A B C D\) is a square, find the coordinates of \(D\) and the length of \(A D\).

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 \(D\).

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 \(\overrightarrow { B C }\). \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\)

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 \(B A C\).

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.

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 \(B P C\).

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

1. Relative to an origin \(O\), the position vectors of two points \(P\) and \(Q\) are \(\mathbf { p }\) and \(\mathbf { q }\) respectively. The point \(R\) is such that \(P Q R\) is a straight line with \(Q\) the mid-point of \(P R\). Find the position vector of \(R\) in terms of \(\mathbf { p }\) and \(\mathbf { q }\), simplifying your answer.
2. The vector \(6 \mathbf { i } + a \mathbf { j } + b \mathbf { k }\) has magnitude 21 and is perpendicular to \(3 \mathbf { i } + 2 \mathbf { j } + 2 \mathbf { k }\). Find the possible values of \(a\) and \(b\), showing all necessary working.

6 With respect to the origin \(O\), the points \(A , B\) and \(C\) have position vectors given by $$\overrightarrow { O A } = \mathbf { i } - \mathbf { k } , \quad \overrightarrow { O B } = 3 \mathbf { i } + 2 \mathbf { j } - 3 \mathbf { k } \quad \text { and } \quad \overrightarrow { O C } = 4 \mathbf { i } - 3 \mathbf { j } + 2 \mathbf { k }$$ The mid-point of \(A B\) is \(M\). The point \(N\) lies on \(A C\) between \(A\) and \(C\) and is such that \(A N = 2 N C\).

1. Find a vector equation of the line \(M N\).
2. It is given that \(M N\) intersects \(B C\) at the point \(P\). Find the position vector of \(P\).

7 With respect to the origin \(O\), the position vectors of two points \(A\) and \(B\) are given by \(\overrightarrow { O A } = \mathbf { i } + 2 \mathbf { j } + 2 \mathbf { k }\) and \(\overrightarrow { O B } = 3 \mathbf { i } + 4 \mathbf { j }\). The point \(P\) lies on the line through \(A\) and \(B\), and \(\overrightarrow { A P } = \lambda \overrightarrow { A B }\).

1. Show that \(\overrightarrow { O P } = ( 1 + 2 \lambda ) \mathbf { i } + ( 2 + 2 \lambda ) \mathbf { j } + ( 2 - 2 \lambda ) \mathbf { k }\).
2. By equating expressions for \(\cos A O P\) and \(\cos B O P\) in terms of \(\lambda\), find the value of \(\lambda\) for which \(O P\) bisects the angle \(A O B\).
3. When \(\lambda\) has this value, verify that \(A P : P B = O A : O B\).

7 The points \(A , B\) and \(C\) have position vectors, relative to the origin \(O\), given by $$\overrightarrow { O A } = \left( \begin{array} { l } 1 \\ 2 \\ 0 \end{array} \right) , \quad \overrightarrow { O B } = \left( \begin{array} { l } 3 \\ 0 \\ 1 \end{array} \right) \quad \text { and } \quad \overrightarrow { O C } = \left( \begin{array} { l } 1 \\ 1 \\ 4 \end{array} \right)$$ The plane \(m\) is perpendicular to \(A B\) and contains the point \(C\).

1. Find a vector equation for the line passing through \(A\) and \(B\).
2. Obtain the equation of the plane \(m\), giving your answer in the form \(a x + b y + c z = d\).
3. The line through \(A\) and \(B\) intersects the plane \(m\) at the point \(N\). Find the position vector of \(N\) and show that \(C N = \sqrt { } ( 13 )\).

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.

9 \includegraphics[max width=\textwidth, alt={}, center]{c1fbc9ef-2dc6-43c3-bc58-179f683c9acf-16_696_1104_264_518} In the diagram, \(O A B C D E F G\) is a cuboid in which \(O A = 2\) units, \(O C = 4\) units and \(O G = 2\) units. Unit vectors \(\mathbf { i } , \mathbf { j }\) and \(\mathbf { k }\) are parallel to \(O A , O C\) and \(O G\) respectively. The point \(M\) is the midpoint of \(D F\). The point \(N\) on \(A B\) is such that \(A N = 3 N B\).

1. Express the vectors \(\overrightarrow { O M }\) and \(\overrightarrow { M N }\) in terms of \(\mathbf { i } , \mathbf { j }\) and \(\mathbf { k }\).
2. Find a vector equation for the line through \(M\) and \(N\).
3. Show that the length of the perpendicular from \(O\) to the line through \(M\) and \(N\) is \(\sqrt { \frac { 53 } { 6 } }\).

8 The points \(A , B\) and \(C\) have position vectors \(\overrightarrow { \mathrm { OA } } = - 2 \mathbf { i } + \mathbf { j } + 4 \mathbf { k } , \overrightarrow { \mathrm { OB } } = 5 \mathbf { i } + 2 \mathbf { j }\) and \(\overrightarrow { \mathrm { OC } } = 8 \mathbf { i } + 5 \mathbf { j } - 3 \mathbf { k }\), where \(O\) is the origin. The line \(l _ { 1 }\) passes through \(B\) and \(C\).

1. Find a vector equation for \(l _ { 1 }\).
   The line \(l _ { 2 }\) has equation \(\mathbf { r } = - 2 \mathbf { i } + \mathbf { j } + 4 \mathbf { k } + \mu ( 3 \mathbf { i } + \mathbf { j } - 2 \mathbf { k } )\).
2. Find the coordinates of the point of intersection of \(l _ { 1 }\) and \(l _ { 2 }\).
3. The point \(D\) on \(l _ { 2 }\) is such that \(\mathrm { AB } = \mathrm { BD }\). Find the position vector of \(D\). \includegraphics[max width=\textwidth, alt={}, center]{5eb2657c-ed74-4ed2-b8c4-08e9e0f657b5-13_58_1545_388_349}

9 Let \(\mathrm { f } ( x ) = \frac { 2 + 11 x - 10 x ^ { 2 } } { ( 1 + 2 x ) ( 1 - 2 x ) ( 2 + x ) }\).

1. Express \(\mathrm { f } ( x )\) in partial fractions.
2. Hence obtain the expansion of \(\mathrm { f } ( x )\) in ascending powers of \(x\), up to and including the term in \(x ^ { 2 }\).

10 With respect to the origin \(O\), the position vectors of the points \(A\) and \(B\) are given by \(\overrightarrow { O A } = \left( \begin{array} { r } 1 \\ 2 \\ - 1 \end{array} \right)\) and \(\overrightarrow { O B } = \left( \begin{array} { l } 0 \\ 3 \\ 1 \end{array} \right)\).

1. Find a vector equation for the line \(l\) through \(A\) and \(B\).
2. The point \(C\) lies on \(l\) and is such that \(\overrightarrow { A C } = 3 \overrightarrow { A B }\). Find the position vector of \(C\).
3. Find the possible position vectors of the point \(P\) on \(l\) such that \(O P = \sqrt { 14 }\).

9 With respect to the origin \(O\), the position vectors of the points \(A , B\) and \(C\) are given by $$\overrightarrow { O A } = \left( \begin{array} { l } 0 \\ 5 \\ 2 \end{array} \right) , \quad \overrightarrow { O B } = \left( \begin{array} { l } 1 \\ 0 \\ 1 \end{array} \right) \quad \text { and } \quad \overrightarrow { O C } = \left( \begin{array} { r } 4 \\ - 3 \\ - 2 \end{array} \right)$$ The midpoint of \(A C\) is \(M\) and the point \(N\) lies on \(B C\), between \(B\) and \(C\), and is such that \(B N = 2 N C\).

1. Find the position vectors of \(M\) and \(N\).
2. Find a vector equation for the line through \(M\) and \(N\).
3. Find the position vector of the point \(Q\) where the line through \(M\) and \(N\) intersects the line through \(A\) and \(B\).

11 \includegraphics[max width=\textwidth, alt={}, center]{ce3c4a9c-bf83-4d28-96e2-ef31c3673dea-16_593_780_264_685} In the diagram, \(O A B C D E F G\) is a cuboid in which \(O A = 3\) units, \(O C = 2\) units and \(O D = 2\) units. Unit vectors \(\mathbf { i } , \mathbf { j }\) and \(\mathbf { k }\) are parallel to \(O A , O D\) and \(O C\) respectively. \(M\) is the midpoint of \(E F\).

1. Find the position vector of \(M\).
   The position vector of \(P\) is \(\mathbf { i } + \mathbf { j } + 2 \mathbf { k }\).
2. Calculate angle PAM.
3. Find the exact length of the perpendicular from \(P\) to the line passing through \(O\) and \(M\).
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

10. With respect to a fixed origin \(O\), the lines \(l _ { 1 }\) and \(l _ { 2 }\) are given by the equations $$\begin{aligned} & l _ { 1 } : \mathbf { r } = ( \mathbf { i } + 5 \mathbf { j } + 5 \mathbf { k } ) + \lambda ( 2 \mathbf { i } + \mathbf { j } - \mathbf { k } ) \\ & l _ { 2 } : \mathbf { r } = ( 2 \mathbf { j } + 12 \mathbf { k } ) + \mu ( 3 \mathbf { i } - \mathbf { j } + 5 \mathbf { k } ) \end{aligned}$$ where \(\lambda\) and \(\mu\) are scalar parameters.

1. Show that \(l _ { 1 }\) and \(l _ { 2 }\) meet and find the position vector of their point of intersection.
2. Show that \(l _ { 1 }\) and \(l _ { 2 }\) are perpendicular to each other. The point \(A\), with position vector \(5 \mathbf { i } + 7 \mathbf { j } + 3 \mathbf { k }\), lies on \(l _ { 1 }\) The point \(B\) is the image of \(A\) after reflection in the line \(l _ { 2 }\)
3. Find the position vector of \(B\). \includegraphics[max width=\textwidth, alt={}, center]{5b698944-41ac-4072-b5e1-c580b7752c39-35_133_163_2604_1786}