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

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 \(A P : P B = O A : O B\).

8 A plane has equation \(4 x - y + 5 z = 39\). A straight line is parallel to the vector \(\mathbf { i } - 3 \mathbf { j } + 4 \mathbf { k }\) and passes through the point \(A ( 0,2 , - 8 )\). The line meets the plane at the point \(B\).

1. Find the coordinates of \(B\).
2. Find the acute angle between the line and the plane.
3. The point \(C\) lies on the line and is such that the distance between \(C\) and \(B\) is twice the distance between \(A\) and \(B\). Find the coordinates of each of the possible positions of the point \(C\).

10 The line \(l\) has vector equation \(\mathbf { r } = \mathbf { i } + 2 \mathbf { j } + \mathbf { k } + \lambda ( 2 \mathbf { i } - \mathbf { j } + \mathbf { k } )\).

1. Find the position vectors of the two points on the line whose distance from the origin is \(\sqrt { } ( 10 )\).
2. The plane \(p\) has equation \(a x + y + z = 5\), where \(a\) is a constant. The acute angle between the line \(l\) and the plane \(p\) is equal to \(\sin ^ { - 1 } \left( \frac { 2 } { 3 } \right)\). Find the possible values of \(a\).

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 \(C N = \sqrt { } ( 13 )\).

7 The points \(A , B , C\) have position vectors $$2 \mathbf { i } + 2 \mathbf { j } , \quad - \mathbf { j } + \mathbf { k } \quad \text { and } \quad 2 \mathbf { i } + \mathbf { j } - 7 \mathbf { k }$$ respectively, relative to the origin \(O\).

1. Find an equation of the plane \(O A B\), giving your answer in the form \(\mathbf { r } . \mathbf { n } = p\).
   The plane \(\Pi\) has equation \(\mathrm { x } - 3 \mathrm { y } - 2 \mathrm { z } = 1\).
2. Find the perpendicular distance of \(\Pi\) from the origin.
3. Find the acute angle between the planes \(O A B\) and \(\Pi\).
4. Find an equation for the common perpendicular to the lines \(O C\) and \(A B\).
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

4 The plane \(\Pi\) contains the lines \(\mathbf { r } = 3 \mathbf { i } - 2 \mathbf { j } + \mathbf { k } + \lambda ( - \mathbf { i } + 2 \mathbf { j } + \mathbf { k } )\) and \(\mathbf { r } = 4 \mathbf { i } + 4 \mathbf { j } + 2 \mathbf { k } + \mu ( 3 \mathbf { i } + 2 \mathbf { j } - \mathbf { k } )\).

1. Find a Cartesian equation of \(\Pi\), giving your answer in the form \(a x + b y + c z = d\).
   The line \(l\) passes through the point \(P\) with position vector \(2 \mathbf { i } + 3 \mathbf { j } + \mathbf { k }\) and is parallel to the vector \(\mathbf { j } + \mathbf { k }\).
2. Find the acute angle between \(I\) and \(\Pi\).
3. Find the position vector of the foot of the perpendicular from \(P\) to \(\Pi\).

6 The lines \(l _ { 1 }\) and \(l _ { 2 }\) have equations \(\mathbf { r } = 2 \mathbf { i } + \mathbf { k } + \lambda ( \mathbf { i } - \mathbf { j } + 2 \mathbf { k } )\) and \(\mathbf { r } = 2 \mathbf { j } + 6 \mathbf { k } + \mu ( \mathbf { i } + 2 \mathbf { j } - 2 \mathbf { k } )\) respectively. The point \(P\) on \(l _ { 1 }\) and the point \(Q\) on \(l _ { 2 }\) are such that \(P Q\) is perpendicular to both \(l _ { 1 }\) and \(l _ { 2 }\).

1. Find the length \(P Q\).
   The plane \(\Pi _ { 1 }\) contains \(P Q\) and \(l _ { 1 }\).
   The plane \(\Pi _ { 2 }\) contains \(P Q\) and \(l _ { 2 }\).
2. 1. Write down an equation of \(\Pi _ { 1 }\), giving your answer in the form \(\mathbf { r } = \mathbf { a } + \mathbf { s b } + \mathbf { t c }\).
   2. Find an equation of \(\Pi _ { 2 }\), giving your answer in the form \(a x + b y + c z = d\).
3. Find the acute angle between \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\).

4 The lines \(l _ { 1 }\) and \(l _ { 2 }\) have equations $$\mathbf { r } = - 2 \mathbf { i } - 3 \mathbf { j } - 5 \mathbf { k } + \lambda ( - 4 \mathbf { i } + 3 \mathbf { j } + 5 \mathbf { k } ) \quad \text { and } \quad \mathbf { r } = 2 \mathbf { i } - 2 \mathbf { j } + 3 \mathbf { k } + \mu ( 2 \mathbf { i } - 3 \mathbf { j } + \mathbf { k } )$$ respectively.

1. Find the shortest distance between \(l _ { 1 }\) and \(l _ { 2 }\).
   The plane \(\Pi\) contains \(l _ { 1 }\) and the point with position vector \(- \mathbf { i } - 3 \mathbf { j } - 4 \mathbf { k }\).
2. Find an equation of \(\Pi\), giving your answer in the form \(a x + b y + c z = d\).

11 With respect to the origin \(O\), the points \(A\) and \(B\) have position vectors given by \(\overrightarrow { O A } = 2 \mathbf { i } - \mathbf { j }\) and \(\overrightarrow { O B } = \mathbf { j } - 2 \mathbf { k }\).

1. Show that \(O A = O B\) and use a scalar product to calculate angle \(A O B\) in degrees.
   The midpoint of \(A B\) is \(M\). The point \(P\) on the line through \(O\) and \(M\) is such that \(P A : O A = \sqrt { 7 } : 1\).
2. Find the possible position vectors of \(P\).
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

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 \(\sqrt { \frac { 53 } { 6 } }\).

9 With respect to the origin \(O\), the point \(A\) has position vector given by \(\overrightarrow { O A } = \mathbf { i } + 5 \mathbf { j } + 6 \mathbf { k }\). The line \(l\) has vector equation \(\mathbf { r } = 4 \mathbf { i } + \mathbf { k } + \lambda ( - \mathbf { i } + 2 \mathbf { j } + 3 \mathbf { k } )\).

1. Find in degrees the acute angle between the directions of \(O A\) and \(l\).
2. Find the position vector of the foot of the perpendicular from \(A\) to \(l\).
3. Hence find the position vector of the reflection of \(A\) in \(l\).

6 Relative to the origin \(O\), the points \(A , B\) and \(C\) have position vectors given by $$\overrightarrow { O A } = \left( \begin{array} { l } 2 \\ 1 \\ 3 \end{array} \right) , \quad \overrightarrow { O B } = \left( \begin{array} { l } 4 \\ 3 \\ 2 \end{array} \right) \quad \text { and } \quad \overrightarrow { O C } = \left( \begin{array} { r } 3 \\ - 2 \\ - 4 \end{array} \right) .$$ The quadrilateral \(A B C D\) is a parallelogram.

1. Find the position vector of \(D\).
2. The angle between \(B A\) and \(B C\) is \(\theta\). Find the exact value of \(\cos \theta\).
3. Hence find the area of \(A B C D\), giving your answer in the form \(p \sqrt { q }\), where \(p\) and \(q\) are integers.

11 The points \(A\) and \(B\) have position vectors \(\mathbf { i } + 2 \mathbf { j } - 2 \mathbf { k }\) and \(2 \mathbf { i } - \mathbf { j } + \mathbf { k }\) respectively. The line \(l\) has equation \(\mathbf { r } = \mathbf { i } - \mathbf { j } + 3 \mathbf { k } + \mu ( 2 \mathbf { i } - 3 \mathbf { j } + 4 \mathbf { k } )\).

1. Show that \(l\) does not intersect the line passing through \(A\) and \(B\).
2. Find the position vector of the foot of the perpendicular from \(A\) to \(l\).
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

9 The lines \(l\) and \(m\) have equations $$\begin{aligned} l : & \mathbf { r } = a \mathbf { i } + 3 \mathbf { j } + b \mathbf { k } + \lambda ( c \mathbf { i } - 2 \mathbf { j } + 4 \mathbf { k } ) \\ m : & \mathbf { r } = \mathbf { i } + 2 \mathbf { j } + 3 \mathbf { k } + \mu ( 2 \mathbf { i } - 3 \mathbf { j } + \mathbf { k } ) \end{aligned}$$ Relative to the origin \(O\), the position vector of the point \(P\) is \(4 \mathbf { i } + 7 \mathbf { j } - 2 \mathbf { k }\).

1. Given that \(l\) is perpendicular to \(m\) and that \(P\) lies on \(l\), find the values of the constants \(a , b\) and \(c\). [4]
2. The perpendicular from \(P\) meets line \(m\) at \(Q\). The point \(R\) lies on \(P Q\) extended, with \(P Q : Q R = 2 : 3\). Find the position vector of \(R\).

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 points \(A , B , C\) and \(D\) have position vectors given by $$\overrightarrow { O A } = \left( \begin{array} { r } 3 \\ - 1 \\ 2 \end{array} \right) , \quad \overrightarrow { O B } = \left( \begin{array} { r } 1 \\ 2 \\ - 3 \end{array} \right) , \quad \overrightarrow { O C } = \left( \begin{array} { r } 1 \\ - 2 \\ 5 \end{array} \right) \quad \text { and } \quad \overrightarrow { O D } = \left( \begin{array} { r } 5 \\ - 6 \\ 11 \end{array} \right)$$

1. Find the obtuse angle between the vectors \(\overrightarrow { O A }\) and \(\overrightarrow { O B }\). \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) The line \(l\) passes through the points \(A\) and \(B\).
2. Find a vector equation for the line \(l\). \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\)
3. Find the position vector of the point of intersection of the line \(l\) and the line passing through \(C\) and \(D\). \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\)

9 Relative to the origin \(O\), the position vectors of the points \(A , B\) and \(C\) are given by $$\overrightarrow { \mathrm { OA } } = 5 \mathbf { i } - 2 \mathbf { j } + \mathbf { k } , \quad \overrightarrow { \mathrm { OB } } = 8 \mathbf { i } + 2 \mathbf { j } - 6 \mathbf { k } \quad \text { and } \quad \overrightarrow { \mathrm { OC } } = 3 \mathbf { i } + 4 \mathbf { j } - 7 \mathbf { k }$$

1. Show that \(O A B C\) is a rectangle. \includegraphics[max width=\textwidth, alt={}, center]{446573d3-73b1-482a-a3f6-1abddfdd90d0-14_67_1573_557_324} \includegraphics[max width=\textwidth, alt={}, center]{446573d3-73b1-482a-a3f6-1abddfdd90d0-14_68_1575_648_322} \includegraphics[max width=\textwidth, alt={}]{446573d3-73b1-482a-a3f6-1abddfdd90d0-14_70_1573_737_324} ....................................................................................................................................... .........................................................................................................................................
2. Use a scalar product to find the acute angle between the diagonals of \(O A B C\).

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 \(O P = \sqrt { 14 }\).

8 \includegraphics[max width=\textwidth, alt={}, center]{bbe57fc0-a8a5-4fe5-a637-4f9db00bdc13-10_588_789_260_678} In the diagram, \(O A B C D\) is a pyramid with vertex \(D\). The horizontal base \(O A B C\) is a square of side 4 units. The edge \(O D\) is vertical and \(O D = 4\) units. The unit vectors \(\mathbf { i } , \mathbf { j }\) and \(\mathbf { k }\) are parallel to \(O A , O C\) and \(O D\) respectively. The midpoint of \(A B\) is \(M\) and the point \(N\) on \(C D\) is such that \(D N = 3 N C\).

1. Find a vector equation for the line through \(M\) and \(N\).
2. Show that the length of the perpendicular from \(O\) to \(M N\) is \(\frac { 1 } { 3 } \sqrt { 82 }\). \(9 \quad\) Let \(\mathrm { f } ( x ) = \frac { 1 } { ( 9 - x ) \sqrt { x } }\).

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}

1. \(A B C D\) is a parallelogram with \(A B\) parallel to \(D C\) and \(A D\) parallel to \(B C\). The position vectors of \(A , B , C\), and \(D\) relative to a fixed origin \(O\) are \(\mathbf { a } , \mathbf { b } , \mathbf { c }\) and \(\mathbf { d }\) respectively.

Given that $$\mathbf { a } = \mathbf { i } + \mathbf { j } - 2 \mathbf { k } , \quad \mathbf { b } = 3 \mathbf { i } - \mathbf { j } + 6 \mathbf { k } , \quad \mathbf { c } = - \mathbf { i } + 3 \mathbf { j } + 6 \mathbf { k }$$

1. find the position vector \(\mathbf { d }\),
2. find the angle between the sides \(A B\) and \(B C\) of the parallelogram,
3. find the area of the parallelogram \(A B C D\). The point \(E\) lies on the line through the points \(C\) and \(D\), so that \(D\) is the midpoint of \(C E\).
4. Use your answer to part (c) to find the area of the trapezium \(A B C E\).