1.10c Magnitude and direction: of vectors

5.The point \(A\) has position vector \(7 \mathbf { i } + 2 \mathbf { j } - 7 \mathbf { k }\) and the point \(B\) has position vector \(12 \mathbf { i } + 3 \mathbf { j } - 15 \mathbf { k }\) .

1. Find a vector for the line \(L _ { 1 }\) which passes through \(A\) and \(B\) . The line \(L _ { 2 }\) has vector equation $$\mathbf { r } = - 4 \mathbf { i } + 12 \mathbf { k } + \mu ( \mathbf { i } - 3 \mathbf { k } )$$
2. Show that \(L _ { 2 }\) passes through the origin \(O\) .
3. Show that \(L _ { 1 }\) and \(L _ { 2 }\) intersect at a point \(C\) and find the position vector of \(C\) .
4. Find the cosine of \(\angle O C A\) .
5. Hence,or otherwise,find the shortest distance from \(O\) to \(L _ { 1 }\) .
6. Show that \(| \overrightarrow { C O } | = | \overrightarrow { A B } |\) .
7. Find a vector equation for the line which bisects \(\angle O C A\) . \includegraphics[max width=\textwidth, alt={}, center]{f9d3e02c-cef2-435b-9cda-76c43fcac575-4_922_1054_279_586} Figure 1 shows a sketch of part of the curve with equation \(y = \mathrm { f } ( x )\), where $$\mathrm { f } ( x ) = x \left( 12 - x ^ { 2 } \right) .$$ The curve cuts the \(x\)-axis at the points \(P , O\) and \(R\), and \(Q\) is the maximum point.

5.The lines \(L _ { 1 }\) and \(L _ { 2 }\) have vector equations \(L _ { 1 } : \quad \mathbf { r } = - 2 \mathbf { i } + 11.5 \mathbf { j } + \lambda ( 3 \mathbf { i } - 4 \mathbf { j } - \mathbf { k } )\), \(L _ { 2 } : \quad \mathbf { r } = 11.5 \mathbf { i } + 3 \mathbf { j } + 8.5 \mathbf { k } + \mu ( 7 \mathbf { i } + 8 \mathbf { j } - 11 \mathbf { k } )\),
where \(\lambda\) and \(\mu\) are parameters.

1. Show that \(L _ { 1 }\) and \(L _ { 2 }\) do not intersect.
2. Show that the vector \(( 2 \mathbf { i } + \mathbf { j } + 2 \mathbf { k } )\) is perpendicular to both \(L _ { 1 }\) and \(L _ { 2 }\) . The point \(A\) lies on \(L _ { 1 }\) ,the point \(B\) lies on \(L _ { 2 }\) and \(A B\) is perpendicular to both \(L _ { 1 }\) and \(L _ { 2 }\) .
3. Find the position vector of the point \(A\) and the position vector of the point \(B\) .
   (8) \includegraphics[max width=\textwidth, alt={}, center]{0df09d8a-7478-4679-b117-128ee226db6a-4_554_1017_404_571} Figure 1 shows a sketch of part of the curve \(C\) with equation $$y = \sin ( \ln x ) , \quad x \geq 1 .$$ The point \(Q\) ,on \(C\) ,is a maximum.

7.The points \(O , P\) and \(Q\) lie on a circle \(C\) with diameter \(O Q\) .The position vectors of \(P\) and \(Q\) , relative to \(O\) ,are \(\mathbf { p }\) and \(\mathbf { q }\) respectively.

1. Prove that \(\mathbf { p } . \mathbf { q } = | \mathbf { p } | ^ { 2 }\) . \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{f2290882-b9a4-43ec-a38f-c44d46477242-6_615_714_412_689} \captionsetup{labelformat=empty} \caption{Figure 3}
   \end{figure} The point \(R\) also lies on \(C\) and \(O P Q R\) is a kite \(K\) as shown in Figure 3.The point \(S\) has position vector,relative to \(O\) ,of \(\lambda \mathbf { q }\) ,where \(\lambda\) is a constant.Given that \(\mathbf { p } = \mathbf { i } + 2 \mathbf { j } - \mathbf { k } , \mathbf { q } = 2 \mathbf { i } + \mathbf { j } - 2 \mathbf { k }\) and that \(O Q\) is perpendicular to \(P S\) ,find
2. the value of \(\lambda\) ,
3. the position vector of \(R\) ,
4. the area of \(K\) . Another circle \(C _ { 1 }\) is drawn inside \(K\) so that the 4 sides of the kite are each tangents to \(C _ { 1 }\) .
5. Find the radius of \(C _ { 1 }\) giving your answer in the form \(( \sqrt { } 2 - 1 ) \sqrt { } n\) ,where \(n\) is an integer. A second kite \(K _ { 1 }\) is similar to \(K\) and is drawn inside \(C _ { 1 }\) .
6. Find that area of \(K _ { 1 }\) .

7.Relative to a fixed origin \(O\) the points \(A , B\) and \(C\) have position vectors $$\mathbf { a } = - \mathbf { i } + \frac { 4 } { 3 } \mathbf { j } + 7 \mathbf { k } , \quad \mathbf { b } = 4 \mathbf { i } + \frac { 4 } { 3 } \mathbf { j } + 2 \mathbf { k } \text { and } \mathbf { c } = 6 \mathbf { i } + \frac { 16 } { 3 } \mathbf { j } + 2 \mathbf { k } \text { respectively. }$$

1. Find the cosine of angle \(A B C\) . The quadrilateral \(A B C D\) is a kite \(K\) .
2. Find the area of \(K\) . A circle is drawn inside \(K\) so that it touches each of the 4 sides of \(K\) .
3. Find the radius of the circle,giving your answer in the form \(p \sqrt { } ( q ) - q \sqrt { } ( p )\) ,where \(p\) and \(q\) are positive integers.
4. Find the position vector of the point \(D\) .
   (Total 18 marks)

11. With respect to a fixed origin \(O\), the lines \(l _ { 1 }\) and \(l _ { 2 }\) are given by the equations $$\begin{aligned} & l _ { 1 } : \mathbf { r } = \left( \begin{array} { l } 7 \\ 4 \\ 9 \end{array} \right) + \lambda \left( \begin{array} { l } 1 \\ 1 \\ 4 \end{array} \right) \\ & l _ { 2 } : \mathbf { r } = \left( \begin{array} { r } - 6 \\ - 7 \\ 3 \end{array} \right) + \mu \left( \begin{array} { l } 5 \\ 4 \\ b \end{array} \right) \end{aligned}$$ where \(\lambda\) and \(\mu\) are scalar parameters and \(b\) is a constant.
Given that \(l _ { 1 }\) and \(l _ { 2 }\) meet at the point \(X\),

1. show that \(b = - 3\) and find the coordinates of \(X\). The point \(A\) lies on \(l _ { 1 }\) and has coordinates (6, 3, 5)
   The point \(B\) lies on \(l _ { 2 }\) and has coordinates \(( 14,9 , - 9 )\)
2. Show that angle \(A X B = \arccos \left( - \frac { 1 } { 10 } \right)\)
3. Using the result obtained in part (b), find the exact area of triangle \(A X B\). Write your answer in the form \(p \sqrt { q }\) where \(p\) and \(q\) are integers to be determined.

3. \begin{figure}[h] \end{figure} Figure 1 shows a regular pentagon \(O A B C D\). The vectors \(\mathbf { p }\) and \(\mathbf { q }\) are defined by \(\mathbf { p } = \overrightarrow { O A }\) and \(\mathbf { q } = \overrightarrow { O D }\) respectively. Let \(k\) be the number such that \(\overrightarrow { D B } = k \overrightarrow { O A }\).

1. Write down \(\overrightarrow { A C }\) in terms of \(\mathbf { p } , \mathbf { q }\) and \(k\) as appropriate.
2. Show that \(\overrightarrow { C D } = - \mathbf { p } - \frac { 1 } { k } \mathbf { q }\)
3. Hence find the value of \(k\) By considering triangle \(D B C\), or otherwise,
4. find the exact value of \(\sin 54 ^ { \circ }\)

6 The plane \(\Pi\) has equation \(\mathbf { r } = \left( \begin{array} { l } 1 \\ 6 \\ 7 \end{array} \right) + \lambda \left( \begin{array} { r } 2 \\ - 1 \\ - 1 \end{array} \right) + \mu \left( \begin{array} { r } 2 \\ - 3 \\ - 5 \end{array} \right)\) and the line \(l\) has equation \(\mathbf { r } = \left( \begin{array} { l } 7 \\ 4 \\ 1 \end{array} \right) + t \left( \begin{array} { r } 3 \\ 0 \\ - 1 \end{array} \right)\).

1. Express the equation of \(\Pi\) in the form r.n \(= p\).
2. Find the point of intersection of \(l\) and \(\Pi\).
3. The equation of \(\Pi\) may be expressed in the form \(\mathbf { r } = \left( \begin{array} { l } 1 \\ 6 \\ 7 \end{array} \right) + \lambda \left( \begin{array} { r } 2 \\ - 1 \\ - 1 \end{array} \right) + \mu \mathbf { c }\), where \(\mathbf { c }\) is perpendicular to \(\left( \begin{array} { r } 2 \\ - 1 \\ - 1 \end{array} \right)\). Find \(\mathbf { c }\).

7

1. Show that the straight line with equation \(\mathbf { r } = \left( \begin{array} { r } 2 \\ - 3 \\ 5 \end{array} \right) + t \left( \begin{array} { r } 1 \\ 4 \\ - 2 \end{array} \right)\) meets the line passing through ( \(9,7,5\) ) and ( \(7,8,2\) ), and find the point of intersection of these lines.
2. Find the acute angle between these lines.

2 Points \(A , B\) and \(C\) have position vectors \(- 5 \mathbf { i } - 10 \mathbf { j } + 12 \mathbf { k } , \mathbf { i } + 2 \mathbf { j } - 3 \mathbf { k }\) and \(3 \mathbf { i } + 6 \mathbf { j } + p \mathbf { k }\) respectively, where \(p\) is a constant.

1. Given that angle \(A B C = 90 ^ { \circ }\), find the value of \(p\).
2. Given instead that \(A B C\) is a straight line, find the value of \(p\).

9 The equation of a straight line \(l\) is \(\mathbf { r } = \left( \begin{array} { l } 3 \\ 1 \\ 1 \end{array} \right) + t \left( \begin{array} { r } 1 \\ - 1 \\ 2 \end{array} \right) . O\) is the origin.

1. The point \(P\) on \(l\) is given by \(t = 1\). Calculate the acute angle between \(O P\) and \(l\).
2. Find the position vector of the point \(Q\) on \(l\) such that \(O Q\) is perpendicular to \(l\).
3. Find the length of \(O Q\).

8 The points \(A ( 3,2,1 ) , B ( 5,4 , - 3 ) , C ( 3,17 , - 4 )\) and \(D ( 1,6,3 )\) form a quadrilateral \(A B C D\).

1. Show that \(A B = A D\).
2. Find a vector equation of the line through \(A\) and the mid-point of \(B D\).
3. Show that \(C\) lies on the line found in part (ii).
4. What type of quadrilateral is \(A B C D\) ?

6 Lines \(l _ { 1 }\) and \(l _ { 2 }\) have vector equations $$\mathbf { r } = \mathbf { j } + \mathbf { k } + t ( 2 \mathbf { i } + a \mathbf { j } + \mathbf { k } ) \quad \text { and } \quad \mathbf { r } = 3 \mathbf { i } - \mathbf { k } + s ( 2 \mathbf { i } + 2 \mathbf { j } - 6 \mathbf { k } )$$ respectively, where \(t\) and \(s\) are parameters and \(a\) is a constant.

1. Given that \(l _ { 1 }\) and \(l _ { 2 }\) are perpendicular, find the value of \(a\).
2. Given instead that \(l _ { 1 }\) and \(l _ { 2 }\) intersect, find
   1. the value of \(a\),
   2. the angle between the lines.

2 Find the unit vector in the direction of \(\left( \begin{array} { c } 2 \\ - 3 \\ \sqrt { 12 } \end{array} \right)\).

+ s \left( \begin{array} { l } 3
2
1 \end{array} \right) \quad \text { and } \quad \mathbf { r } = \left( \begin{array} { l } 1
0
0 \end{array} \right) + t \left( \begin{array} { r } 0
1
- 1 \end{array} \right)$$ respectively.\\

1. Show that \(l _ { 1 }\) and \(l _ { 2 }\) are skew.
2. Find the acute angle between \(l _ { 1 }\) and \(l _ { 2 }\).
3. The point \(A\) lies on \(l _ { 1 }\) and \(O A\) is perpendicular to \(l _ { 1 }\), where \(O\) is the origin. Find the position vector of \(A\). 6 Find the coefficient of \(x ^ { 2 }\) in the expansion in ascending powers of \(x\) of $$\sqrt { \frac { 1 + a x } { 4 - x } } ,$$ giving your answer in terms of \(a\).

5 \includegraphics[max width=\textwidth, alt={}, center]{b5d85e48-0d5a-4edf-bf58-eba4f8d28d3d-2_425_680_1302_689} In the diagram the points \(A\) and \(B\) have position vectors \(\mathbf { a }\) and \(\mathbf { b }\) with respect to the origin \(O\). Given that \(| \mathbf { a } | = 3 , | \mathbf { b } | = 4\) and \(\mathbf { a . b } = 6\), find

1. the angle \(A O B\),
2. \(| \mathbf { a } - \mathbf { b } |\).

7 Points \(A ( 2,2,5 ) , B ( 1 , - 1 , - 4 ) , C ( 3,3,10 )\) and \(D ( 8,6,3 )\) are the vertices of a pyramid with a triangular base.

1. Calculate the lengths \(A B\) and \(A C\), and the angle \(B A C\).
2. Show that \(\overrightarrow { A D }\) is perpendicular to both \(\overrightarrow { A B }\) and \(\overrightarrow { A C }\).
3. Calculate the volume of the pyramid \(A B C D\).
   [0pt] [The volume of the pyramid is \(V = \frac { 1 } { 3 } \times\) base area × perpendicular height.]

2 The points \(O ( 0,0,0 ) , A ( 2,8,2 ) , B ( 5,5,8 )\) and \(C ( 3 , - 3,6 )\) form a parallelogram \(O A B C\). Use a scalar product to find the acute angle between the diagonals of this parallelogram.

8 The points \(A\) and \(B\) have position vectors relative to the origin \(O\) given by $$\overrightarrow { O A } = \left( \begin{array} { c } 3 \sin \alpha \\ 2 \cos \alpha \\ - 1 \end{array} \right) \text { and } \overrightarrow { O B } = \left( \begin{array} { c } 2 \cos \alpha \\ 4 \sin \alpha \\ 3 \end{array} \right)$$ where \(0 ^ { \circ } < \alpha < 90 ^ { \circ }\). It is given that \(\overrightarrow { O A }\) and \(\overrightarrow { O B }\) are perpendicular.

1. Calculate the two possible values of \(\alpha\).
2. Calculate the area of triangle \(O A B\) for the smaller value of \(\alpha\) from part (i).

4 The points A , B and C have coordinates \(( 2,0 , - 1 ) , ( 4,3 , - 6 )\) and \(( 9,3 , - 4 )\) respectively.

1. Show that AB is perpendicular to BC .
2. Find the area of triangle ABC .

7 A tent has vertices ABCDEF with coordinates as shown in Fig. 7. Lengths are in metres. The \(\mathrm { O } x y\) plane is horizontal. \begin{figure}[h] \end{figure}

1. Find the length of the ridge of the tent DE , and the angle this makes with the horizontal.
2. Show that the vector \(\mathbf { i } - 4 \mathbf { j } + 5 \mathbf { k }\) is normal to the plane through \(\mathrm { A } , \mathrm { D }\) and E . Hence find the equation of this plane. Given that B lies in this plane, find \(a\).
3. Verify that the equation of the plane BCD is \(x + z = 8\). Hence find the acute angle between the planes ABDE and BCD .

6 Fig. 6 shows a lean-to greenhouse ABCDHEFG . With respect to coordinate axes Oxyz , the coordinates of the vertices are as shown. All distances are in metres. Ground level is the plane \(z = 0\). \begin{figure}[h] \end{figure}

1. Verify that the equation of the plane through \(\mathrm { A } , \mathrm { B }\) and E is \(x + 6 y + 12 = 0\). Hence, given that F lies in this plane, show that \(a = - 2 \frac { 1 } { 3 }\).
2. (A) Show that the vector \(\left( \begin{array} { r } 1 \\ - 6 \\ 0 \end{array} \right)\) is normal to the plane DHC.
   (B) Hence find the cartesian equation of this plane.
   (C) Given that G lies in the plane DHC , find \(b\) and the length FG .
3. Find the angle EFB . A straight wire joins point H to a point P which is half way between E and F . Q is a point two-thirds of the way along this wire, so that \(\mathrm { HQ } = 2 \mathrm { QP }\).
4. Find the height of Q above the ground. \section*{Question 7 begins on page 4.}

3 The resultant of the force \(\binom { - 4 } { 8 } \mathrm {~N}\) and the force \(\mathbf { F }\) gives an object of mass 6 kg an acceleration of \(\binom { 2 } { 3 } \mathrm {~ms} ^ { - 2 }\).

1. Calculate \(\mathbf { F }\).
2. Calculate the angle between \(\mathbf { F }\) and the vector \(\binom { 0 } { 1 }\).

5 In this question the unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are pointing east and north respectively.

1. Calculate the bearing of the vector \(- 4 \mathbf { i } - 6 \mathbf { j }\). The vector \(- 4 \mathbf { i } - 6 \mathbf { j } + k ( 3 \mathbf { i } - 2 \mathbf { j } )\) is in the direction \(7 \mathbf { i } - 9 \mathbf { j }\).
2. Find \(k\).

3 Force \(\mathbf { F }\) is \(\left( \begin{array} { r } - 2 \\ 3 \\ - 4 \end{array} \right) \mathrm { N }\), force \(\mathbf { G }\) is \(\left( \begin{array} { r } - 6 \\ y \\ z \end{array} \right) \mathrm { N }\) and force \(\mathbf { H }\) is \(\left( \begin{array} { r } 3 \\ - 5 \\ - 1 \end{array} \right) \mathrm { N }\).

1. Given that \(\mathbf { F }\) and \(\mathbf { G }\) act in parallel lines, find \(y\) and \(z\). Forces \(\mathbf { F }\) and \(\mathbf { H }\) are the only forces acting on an object of mass 5 kg .
2. Calculate the acceleration of the object. Calculate also the magnitude of this acceleration.

8 In this question, positions are given relative to a fixed origin, O. The \(x\)-direction is east and the \(y\)-direction north; distances are measured in kilometres. Two boats, the Rosemary and the Sage, are having a race between two points A and B.
The position vector of the Rosemary at time \(t\) hours after the start is given by $$\mathbf { r } = \binom { 3 } { 2 } + \binom { 6 } { 8 } t , \text { where } 0 \leqslant t \leqslant 2 .$$ The Rosemary is at point A when \(t = 0\), and at point B when \(t = 2\).

1. Find the distance AB .
2. Show that the Rosemary travels at constant velocity. The position vector of the Sage is given by $$\mathbf { r } = \binom { 3 ( 2 t + 1 ) } { 2 \left( 2 t ^ { 2 } + 1 \right) }$$
3. Plot the points A and B . Draw the paths of the two boats for \(0 \leqslant t \leqslant 2\).
4. What can you say about the result of the race?
5. Find the speed of the Sage when \(t = 2\). Find also the direction in which it is travelling, giving your answer as a compass bearing, to the nearest degree.
6. Find the displacement of the Rosemary from the Sage at time \(t\) and hence calculate the greatest distance between the boats during the race.