1.10d Vector operations: addition and scalar multiplication

9
The diagram shows three points \(A , B\) and \(C\) whose position vectors with respect to the origin \(O\) are given by \(\overrightarrow { O A } = \left( \begin{array} { r } 2 \\ - 1 \\ 2 \end{array} \right) , \overrightarrow { O B } = \left( \begin{array} { l } 0 \\ 3 \\ 1 \end{array} \right)\) and \(\overrightarrow { O C } = \left( \begin{array} { l } 3 \\ 0 \\ 4 \end{array} \right)\). The point \(D\) lies on \(B C\), between \(B\) and \(C\), and is such that \(C D = 2 D B\).

1. Find the equation of the plane \(A B C\), giving your answer in the form \(a x + b y + c z = d\).
2. Find the position vector of \(D\).
3. Show that the length of the perpendicular from \(A\) to \(O D\) is \(\frac { 1 } { 3 } \sqrt { } ( 65 )\).

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 \(a x + b y + c z = d\).

9 With respect to the origin \(O\), the vertices of a triangle \(A B C\) have position vectors $$\overrightarrow { O A } = 2 \mathbf { i } + 5 \mathbf { k } , \quad \overrightarrow { O B } = 3 \mathbf { i } + 2 \mathbf { j } + 3 \mathbf { k } \quad \text { and } \quad \overrightarrow { O C } = \mathbf { i } + \mathbf { j } + \mathbf { k }$$

1. Using a scalar product, show that angle \(A B C\) is a right angle.
2. Show that triangle \(A B C\) is isosceles.
3. Find the exact length of the perpendicular from \(O\) to the line through \(B\) and \(C\).

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

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.

9 The equations of two straight lines \(l _ { 1 }\) and \(l _ { 2 }\) are $$l _ { 1 } : \quad \mathbf { r } = \mathbf { i } - 2 \mathbf { j } + 3 \mathbf { k } + \lambda ( 2 \mathbf { i } - \mathbf { j } + a \mathbf { k } ) \quad \text { and } \quad l _ { 2 } : \quad \mathbf { r } = - \mathbf { i } - \mathbf { j } - \mathbf { k } + \mu ( 3 \mathbf { i } - 2 \mathbf { j } - 2 \mathbf { k } ) ,$$ where \(a\) is a constant.
The lines \(l _ { 1 }\) and \(l _ { 2 }\) are perpendicular.

1. Show that \(a = 4\).
   The lines \(l _ { 1 }\) and \(l _ { 2 }\) also intersect.
2. Find the position vector of the point of intersection.
   The point \(A\) has position vector \(- 5 \mathbf { i } + \mathbf { j } - 9 \mathbf { k }\).
3. Show that \(A\) lies on \(l _ { 1 }\).
   The point \(B\) is the image of \(A\) after a reflection in the line \(l _ { 2 }\).
4. Find the position vector of \(B\).

8 With respect to the origin \(O\), the position vectors of the points \(A , B , C\) and \(D\) are given by $$\overrightarrow { O A } = \left( \begin{array} { l } 2 \\ 1 \\ 5 \end{array} \right) , \quad \overrightarrow { O B } = \left( \begin{array} { r } 4 \\ - 1 \\ 1 \end{array} \right) , \quad \overrightarrow { O C } = \left( \begin{array} { l } 1 \\ 1 \\ 2 \end{array} \right) \quad \text { and } \quad \overrightarrow { O D } = \left( \begin{array} { l } 3 \\ 2 \\ 3 \end{array} \right)$$

1. Show that \(A B = 2 C D\).
2. Find the angle between the directions of \(\overrightarrow { A B }\) and \(\overrightarrow { C D }\).
3. Show that the line through \(A\) and \(B\) does not intersect the line through \(C\) and \(D\).

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 \(\mathrm { f } ( x ) = \frac { 1 } { ( 9 - x ) \sqrt { x } }\).

10 Let \(\mathrm { f } ( x ) = \frac { 2 x ^ { 2 } + 7 x + 8 } { ( 1 + x ) ( 2 + x ) ^ { 2 } }\).

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 \(x ^ { 2 }\). \includegraphics[max width=\textwidth, alt={}, center]{98001cfe-46a1-4c8f-9230-c140ebff6176-18_737_1034_262_552} In the diagram, \(O A B C D\) is a solid figure in which \(O A = O B = 4\) units and \(O D = 3\) units. The edge \(O D\) is vertical, \(D C\) is parallel to \(O B\) and \(D C = 1\) unit. The base, \(O A B\), is horizontal and angle \(A O B = 90 ^ { \circ }\). Unit vectors \(\mathbf { i } , \mathbf { j }\) and \(\mathbf { k }\) are parallel to \(O A , O B\) and \(O D\) respectively. The midpoint of \(A B\) is \(M\) and the point \(N\) on \(B C\) is such that \(C N = 2 N B\).
   1. Express vectors \(\overrightarrow { M D }\) and \(\overrightarrow { O N }\) in terms of \(\mathbf { i } , \mathbf { j }\) and \(\mathbf { k }\).
   2. Calculate the angle in degrees between the directions of \(\overrightarrow { M D }\) and \(\overrightarrow { O N }\).
   3. Show that the length of the perpendicular from \(M\) to \(O N\) is \(\sqrt { \frac { 22 } { 5 } }\).
      If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

2 \includegraphics[max width=\textwidth, alt={}, center]{f7a22c07-44e3-4891-be60-cbab772f45df-2_549_589_934_778} Two forces, each of magnitude 8 N , act at a point in the directions \(O A\) and \(O B\). The angle between the forces is \(\theta ^ { \circ }\) (see diagram). The resultant of the two forces has component 9 N in the direction \(O A\). Find

1. the value of \(\theta\),
2. the magnitude of the resultant of the two forces.

11. With respect to a fixed origin \(O\) the lines \(l _ { 1 }\) and \(l _ { 2 }\) are given by the equations $$l _ { 1 } : \mathbf { r } = \left( \begin{array} { r } 14 \\ - 6 \\ - 13 \end{array} \right) + \lambda \left( \begin{array} { r } - 2 \\ 1 \\ 4 \end{array} \right) \quad l _ { 2 } : \mathbf { r } = \left( \begin{array} { r } p \\ - 7 \\ 4 \end{array} \right) + \mu \left( \begin{array} { l } q \\ 2 \\ 1 \end{array} \right)$$ where \(\lambda\) and \(\mu\) are scalar parameters and \(p\) and \(q\) are constants. Given that \(l _ { 1 }\) and \(l _ { 2 }\) are perpendicular,

1. show that \(q = 3\) Given further that \(l _ { 1 }\) and \(l _ { 2 }\) intersect at point \(X\), find
2. the value of \(p\),
3. the coordinates of \(X\). The point \(A\) lies on \(l _ { 1 }\) and has position vector \(\left( \begin{array} { r } 6 \\ - 2 \\ 3 \end{array} \right)\) Given that point \(B\) also lies on \(l _ { 1 }\) and that \(A B = 2 A X\)
4. find the two possible position vectors of \(B\). \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \section*{Question 11 continued} \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\)

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 \(A B C E\).

7. With respect to a fixed origin \(O\), the lines \(l _ { 1 }\) and \(l _ { 2 }\) are given by the equations $$\begin{aligned} & l _ { 1 } : \mathbf { r } = ( 13 \mathbf { i } + 15 \mathbf { j } - 8 \mathbf { k } ) + \lambda ( 3 \mathbf { i } + 3 \mathbf { j } - 4 \mathbf { k } ) \\ & l _ { 2 } : \mathbf { r } = ( 7 \mathbf { i } - 6 \mathbf { j } + 14 \mathbf { k } ) + \mu ( 2 \mathbf { i } - 3 \mathbf { j } + 2 \mathbf { k } ) \end{aligned}$$ 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, \(B\).
2. Find the acute angle between the lines \(l _ { 1 }\) and \(l _ { 2 }\) The point \(A\) has position vector \(- 5 \mathbf { i } - 3 \mathbf { j } + 16 \mathbf { k }\)
3. Show that \(A\) lies on \(l _ { 1 }\) The point \(C\) lies on the line \(l _ { 1 }\) where \(\overrightarrow { A B } = \overrightarrow { B C }\)
4. Find the position vector of \(C\).
   
   \section*{"}

6. Relative to a fixed origin \(O\), the points \(A\), \(B\) and \(C\) have coordinates ( \(2,1,9 ) , ( 5,2,7 )\) and \(( 4 , - 3,3 )\) respectively. The line \(l\) passes through the points \(A\) and \(B\).

1. Find a vector equation for the line \(l\).
2. Find, in degrees, the acute angle between the line \(I\) and the line \(A C\). The point \(D\) lies on the line \(l\) such that angle \(A C D\) is \(90 ^ { \circ }\)
3. Find the coordinates of \(D\).
4. Find the exact area of triangle \(A D C\), giving your answer as a fully simplified surd.

14. Relative to a fixed origin \(O\), the line \(l\) has vector equation $$\mathbf { r } = \left( \begin{array} { r } - 1 \\ - 4 \\ 6 \end{array} \right) + \lambda \left( \begin{array} { r } 2 \\ 1 \\ - 1 \end{array} \right)$$ where \(\lambda\) is a scalar parameter. Points \(A\) and \(B\) lie on the line \(l\), where \(A\) has coordinates ( \(1 , a , 5\) ) and \(B\) has coordinates ( \(b , - 1,3\) ).

1. Find the value of the constant \(a\) and the value of the constant \(b\).
2. Find the vector \(\overrightarrow { A B }\). The point \(C\) has coordinates ( \(4 , - 3,2\) )
3. Show that the size of the angle \(C A B\) is \(30 ^ { \circ }\)
4. Find the exact area of the triangle \(C A B\), giving your answer in the form \(k \sqrt { 3 }\), where \(k\) is a constant to be determined. The point \(D\) lies on the line \(l\) so that the area of the triangle \(C A D\) is twice the area of the triangle \(C A B\).
5. Find the coordinates of the two possible positions of \(D\).

8. With respect to a fixed origin \(O\), the lines \(l _ { 1 }\) and \(l _ { 2 }\) are given by the equations $$l _ { 1 } : \mathbf { r } = \left( \begin{array} { r } 1 \\ - 3 \\ 2 \end{array} \right) + \lambda \left( \begin{array} { l } 1 \\ 2 \\ 3 \end{array} \right) , \quad l _ { 2 } : \mathbf { r } = \left( \begin{array} { l } 6 \\ 4 \\ 1 \end{array} \right) + \mu \left( \begin{array} { r } 1 \\ 1 \\ - 1 \end{array} \right)$$ where \(\lambda\) and \(\mu\) are scalar parameters.

1. Show that \(l _ { 1 }\) and \(l _ { 2 }\) do not meet. The point \(P\) is on \(l _ { 1 }\) where \(\lambda = 0\), and the point \(Q\) is on \(l _ { 2 }\) where \(\mu = - 1\)
2. Find the acute angle between the line segment \(P Q\) and \(l _ { 1 }\), giving your answer in degrees to 2 decimal places.
3. Find the shortest distance from the point \(Q\) to the line \(l _ { 1 }\), giving your answer to 3 significant figures.

1. Relative to a fixed origin \(O\),
   the point \(A\) has position vector \(( 2 \mathbf { i } - 3 \mathbf { j } - 2 \mathbf { k } )\) the point \(B\) has position vector \(( 3 \mathbf { i } + 2 \mathbf { j } + 5 \mathbf { k } )\) the point \(C\) has position vector ( \(2 \mathbf { i } + 4 \mathbf { j } - 3 \mathbf { k }\) )

The line \(l\) passes through the points \(A\) and \(B\).

1. Find the vector \(\overrightarrow { A B }\).
2. Find a vector equation for the line \(l\).
3. Show that the size of the angle \(C A B\) is \(62.8 ^ { \circ }\), to one decimal place.
4. Hence find the area of triangle \(C A B\), giving your answer to 3 significant figures. The point \(D\) lies on the line \(l\). Given that the area of triangle \(C A D\) is twice the area of triangle \(C A B\),
5. find the two possible position vectors of point \(D\).
   

11. Relative to a fixed origin \(O\), the line \(l _ { 1 }\) is given by the equation $$l _ { 1 } : \quad \mathbf { r } = \left( \begin{array} { r } 2 \\ 3 \\ - 1 \end{array} \right) + \lambda \left( \begin{array} { r } - 1 \\ 4 \\ 3 \end{array} \right)$$ where \(\lambda\) is a scalar parameter. The line \(l _ { 2 }\) passes through the origin and is parallel to \(l _ { 1 }\)

1. Find a vector equation for \(l _ { 2 }\) The point \(A\) and the point \(B\) both lie on \(l _ { 1 }\) with parameters \(\lambda = 0\) and \(\lambda = 3\) respectively.
   Write down
2. 1. the coordinates of \(A\),
   2. the coordinates of \(B\).
3. Find the size of the acute angle between \(O A\) and \(l _ { 1 }\) Give your answer in degrees to one decimal place. The point \(D\) lies on \(l _ { 2 }\) such that \(O A B D\) is a parallelogram.
4. Find the area of \(O A B D\), giving your answer to the nearest whole number.

1. The line \(l _ { 1 }\) has equation \(\mathbf { r } = \left( \begin{array} { r } 2 \\ 3 \\ - 4 \end{array} \right) + \lambda \left( \begin{array} { l } 1 \\ 2 \\ 1 \end{array} \right)\), where \(\lambda\) is a scalar parameter.

The line \(l _ { 2 }\) has equation \(\mathbf { r } = \left( \begin{array} { r } 0 \\ 9 \\ - 3 \end{array} \right) + \mu \left( \begin{array} { l } 5 \\ 0 \\ 2 \end{array} \right)\), where \(\mu\) is a scalar parameter.
Given that \(l _ { 1 }\) and \(l _ { 2 }\) meet at the point \(C\), find

1. the coordinates of \(C\). The point \(A\) is the point on \(l _ { 1 }\) where \(\lambda = 0\) and the point \(B\) is the point on \(l _ { 2 }\) where \(\mu = - 1\)
2. Find the size of the angle \(A C B\). Give your answer in degrees to 2 decimal places.
3. Hence, or otherwise, find the area of the triangle \(A B C\).

7. The point \(A\) has position vector \(\mathbf { a } = 2 \mathbf { i } + 2 \mathbf { j } + \mathbf { k }\) and the point \(B\) has position vector \(\mathbf { b } = \mathbf { i } + \mathbf { j } - 4 \mathbf { k }\), relative to an origin \(O\).

1. Find the position vector of the point \(C\), with position vector \(\mathbf { c }\), given by $$\mathbf { c } = \mathbf { a } + \mathbf { b } .$$
2. Show that \(O A C B\) is a rectangle, and find its exact area. The diagonals of the rectangle, \(A B\) and \(O C\), meet at the point \(D\).
3. Write down the position vector of the point \(D\).
4. Find the size of the angle \(A D C\).

6. The points \(A\) and \(B\) have position vectors \(2 \mathbf { i } + 6 \mathbf { j } - \mathbf { k }\) and \(3 \mathbf { i } + 4 \mathbf { j } + \mathbf { k }\) respectively. The line \(l _ { 1 }\) passes through the points \(A\) and \(B\).

1. Find the vector \(\overrightarrow { A B }\).
2. Find a vector equation for the line \(l _ { 1 }\). A second line \(l _ { 2 }\) passes through the origin and is parallel to the vector \(\mathbf { i } + \mathbf { k }\). The line \(l _ { 1 }\) meets the line \(l _ { 2 }\) at the point \(C\).
3. Find the acute angle between \(l _ { 1 }\) and \(l _ { 2 }\).
4. Find the position vector of the point \(C\).