1.10e Position vectors: and displacement

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 \(A B C E\).

6. With respect to a fixed origin, the point \(A\) with position vector \(\mathbf { i } + 2 \mathbf { j } + 3 \mathbf { k }\) lies on the line \(l _ { 1 }\) with equation $$\mathbf { r } = \left( \begin{array} { l } 1 \\ 2 \\ 3 \end{array} \right) + \lambda \left( \begin{array} { r } 0 \\ 2 \\ - 1 \end{array} \right) , \quad \text { where } \lambda \text { is a scalar parameter, }$$ and the point \(B\) with position vector \(4 \mathbf { i } + p \mathbf { j } + 3 \mathbf { k }\), where \(p\) is a constant, lies on the line \(l _ { 2 }\) with equation $$\mathbf { r } = \left( \begin{array} { l } 7 \\ 0 \\ 7 \end{array} \right) + \mu \left( \begin{array} { r } 3 \\ - 5 \\ 4 \end{array} \right) , \quad \text { where } \mu \text { is a scalar parameter. }$$

1. Find the value of the constant \(p\).
2. Show that \(l _ { 1 }\) and \(l _ { 2 }\) intersect and find the position vector of their point of intersection, \(C\).
3. Find the size of the angle \(A C B\), giving your answer in degrees to 3 significant figures.
4. Find the area of the triangle \(A B C\), giving your answer to 3 significant figures. \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \section*{Question 6 continued} \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\)

7. The point \(A\) with coordinates ( \(- 3,7,2\) ) lies on a line \(l _ { 1 }\) The point \(B\) also lies on the line \(l _ { 1 }\) Given that \(\quad \overrightarrow { A B } = \left( \begin{array} { r } 4 \\ - 6 \\ 2 \end{array} \right)\),

1. find the coordinates of point \(B\). The point \(P\) has coordinates ( \(9,1,8\) )
2. Find the cosine of the angle \(P A B\), giving your answer as a simplified surd.
3. Find the exact area of triangle \(P A B\), giving your answer in its simplest form. The line \(l _ { 2 }\) passes through the point \(P\) and is parallel to the line \(l _ { 1 }\)
4. Find a vector equation for the line \(l _ { 2 }\) The point \(Q\) lies on the line \(l _ { 2 }\) Given that the line segment \(A P\) is perpendicular to the line segment \(B Q\),
5. find the coordinates of the point \(Q\).

8. Relative to a fixed origin \(O\), the lines \(l _ { 1 }\) and \(l _ { 2 }\) are given by the equations $$\begin{aligned} & l _ { 1 } : \quad \mathbf { r } = \left( \begin{array} { r } 4 \\ - 3 \\ 2 \end{array} \right) + \lambda \left( \begin{array} { r } 3 \\ - 2 \\ - 1 \end{array} \right) \quad \text { where } \lambda \text { is a scalar parameter } \\ & l _ { 2 } : \quad \mathbf { r } = \left( \begin{array} { r } 2 \\ 0 \\ - 9 \end{array} \right) + \mu \left( \begin{array} { r } 2 \\ - 1 \\ - 3 \end{array} \right) \quad \text { where } \mu \text { is a scalar parameter } \end{aligned}$$ Given that \(l _ { 1 }\) and \(l _ { 2 }\) meet at the point \(X\),

1. find the position vector of \(X\). The point \(P ( 10 , - 7,0 )\) lies on \(l _ { 1 }\) The point \(Q\) lies on \(l _ { 2 }\) Given that \(\overrightarrow { P Q }\) is perpendicular to \(l _ { 2 }\)
2. calculate the coordinates of \(Q\).
   

7. [In this question, the unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are due east and due north respectively. Position vectors are relative to a fixed origin \(O\).] A boat \(P\) is moving with constant velocity \(( - 4 \mathbf { i } + 8 \mathbf { j } ) \mathrm { km } \mathrm { h } ^ { - 1 }\).

1. Calculate the speed of \(P\). When \(t = 0\), the boat \(P\) has position vector \(( 2 \mathbf { i } - 8 \mathbf { j } ) \mathrm { km }\). At time \(t\) hours, the position vector of \(P\) is \(\mathbf { p ~ k m }\).
2. Write down \(\mathbf { p }\) in terms of \(t\). A second boat \(Q\) is also moving with constant velocity. At time \(t\) hours, the position vector of \(Q\) is \(\mathbf { q } \mathrm { km }\), where $$\mathbf { q } = 18 \mathbf { i } + 12 \mathbf { j } - t ( 6 \mathbf { i } + 8 \mathbf { j } )$$ Find
3. the value of \(t\) when \(P\) is due west of \(Q\),
4. the distance between \(P\) and \(Q\) when \(P\) is due west of \(Q\).

7. [In this question, the horizontal unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are directed due east and north respectively] A mountain rescue post \(O\) receives a distress call via a mobile phone from a walker who has broken a leg and cannot move. The walker says he is by a pipeline and he can also see a radio mast which he believes to be south-west of him. The pipeline is known to run north-south for a long distance through the point with position vector \(6 \mathbf { i } \mathrm {~km}\), relative to \(O\). The radio mast is known to be at the point with position vector \(2 \mathbf { j } \mathrm {~km}\), relative to \(O\).

1. Using the information supplied by the walker, write down his position vector in the form \(( a \mathbf { i } + b \mathbf { j } )\). The rescue party moves at a horizontal speed of \(5 \mathrm {~km} \mathrm {~h} ^ { - 1 }\). The leader of the party wants to give the walker and idea of how long it will take to for the rescue party to arrive.
2. Calculate how long it will take for the rescue party to reach the walker's estimated position. The rescue party sets out and walks straight towards the walker's estimated position at a constant horizontal speed of \(5 \mathrm {~km} \mathrm {~h} ^ { - 1 }\). After the party has travelled for one hour, the walker rings again. He is very apologetic and says that he now realises that the radio mask is in fact north-west of his position
3. Find the position vector of the walker.
4. Find in degrees to one decimal place, the bearing on which the rescue party should now travel in order to reach the walker directly. \section*{END}

6. [In this question \(\mathbf { i }\) and \(\mathbf { j }\) are horizontal unit vectors due east and due north respectively and position vectors are given with respect to a fixed origin.] A ship \(S\) is moving with constant velocity \(( - 12 \mathbf { i } + 7.5 \mathbf { j } ) \mathrm { km } \mathrm { h } ^ { - 1 }\).

1. Find the direction in which \(S\) is moving, giving your answer as a bearing. At time \(t\) hours after noon, the position vector of \(S\) is \(\mathbf { s } \mathrm { km }\). When \(t = 0 , \mathbf { s } = 40 \mathbf { i } - 6 \mathbf { j }\).
2. Write down \(\mathbf { s }\) in terms of \(t\). A fixed beacon \(B\) is at the point with position vector \(( 7 \mathbf { i } + 12.5 \mathbf { j } ) \mathrm { km }\).
3. Find the distance of \(S\) from \(B\) when \(t = 3\)
4. Find the distance of \(S\) from \(B\) when \(S\) is due north of \(B\).

1. A particle \(P\) is moving with constant velocity. The position vector of \(P\) at time \(t\) seconds \(( t \geqslant 0 )\) is \(\mathbf { r }\) metres, relative to a fixed origin \(O\), and is given by

$$\mathbf { r } = ( 2 t - 3 ) \mathbf { i } + ( 4 - 5 t ) \mathbf { j }$$

1. Find the initial position vector of \(P\). The particle \(P\) passes through the point with position vector \(( 3.4 \mathbf { i } - 12 \mathbf { j } )\) m at time \(T\) seconds.
2. Find the value of \(T\).
3. Find the speed of \(P\).

1. \hspace{0pt} [In this question \(\mathbf { i }\) and \(\mathbf { j }\) are horizontal unit vectors due east and due north respectively and position vectors are given relative to a fixed origin \(O\).]

Two cars \(P\) and \(Q\) are moving on straight horizontal roads with constant velocities. The velocity of \(P\) is \(( 15 \mathbf { i } + 20 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\) and the velocity of \(Q\) is \(( 20 \mathbf { i } - 5 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\)

1. Find the direction of motion of \(Q\), giving your answer as a bearing to the nearest degree. At time \(t = 0\), the position vector of \(P\) is \(400 \mathbf { i }\) metres and the position vector of \(Q\) is 800j metres. At time \(t\) seconds, the position vectors of \(P\) and \(Q\) are \(\mathbf { p }\) metres and \(\mathbf { q }\) metres respectively.
2. Find an expression for
   1. \(\mathbf { p }\) in terms of \(t\),
   2. \(\mathbf { q }\) in terms of \(t\).
3. Find the position vector of \(Q\) when \(Q\) is due west of \(P\).

7. Two helicopters \(P\) and \(Q\) are moving in the same horizontal plane. They are modelled as particles moving in straight lines with constant speeds. At noon \(P\) is at the point with position vector \(( 20 \mathbf { i } + 35 \mathbf { j } ) \mathrm { km }\) with respect to a fixed origin \(O\). At time \(t\) hours after noon the position vector of \(P\) is \(\mathbf { p } \mathrm { km }\). When \(t = \frac { 1 } { 2 }\) the position vector of \(P\) is \(( 50 \mathbf { i } - 25 \mathbf { j } ) \mathrm { km }\). Find

1. the velocity of \(P\) in the form \(( a \mathbf { i } + b \mathbf { j } ) \mathrm { km } \mathrm { h } ^ { - 1 }\),
2. an expression for \(\mathbf { p }\) in terms of \(t\). At noon \(Q\) is at \(O\) and at time \(t\) hours after noon the position vector of \(Q\) is \(\mathbf { q } \mathrm { km }\). The velocity of \(Q\) has magnitude \(120 \mathrm {~km} \mathrm {~h} ^ { - 1 }\) in the direction of \(4 \mathbf { i } - 3 \mathbf { j }\). Find
   (d) an expression for \(\mathbf { q }\) in terms of \(t\),
   (e) the distance, to the nearest km , between \(P\) and \(Q\) when \(t = 2\). \section*{8.} \section*{Figure 4}
   \includegraphics[max width=\textwidth, alt={}]{14703bfa-abd8-4a8d-bc18-20d66eea409e-6_695_1153_322_562}
   Two particles \(A\) and \(B\), of mass \(m \mathrm {~kg}\) and 3 kg respectively, are connected by a light inextensible string. The particle \(A\) is held resting on a smooth fixed plane inclined at \(30 ^ { \circ }\) to the horizontal. The string passes over a smooth pulley \(P\) fixed at the top of the plane. The portion \(A P\) of the string lies along a line of greatest slope of the plane and \(B\) hangs freely from the pulley, as shown in Fig. 4. The system is released from rest with \(B\) at a height of 0.25 m above horizontal ground. Immediately after release, \(B\) descends with an acceleration of \(\frac { 2 } { 5 } g\). Given that \(A\) does not reach \(P\), calculate
   (a) the tension in the string while \(B\) is descending,
   (b) the value of \(m\). The particle \(B\) strikes the ground and does not rebound. Find
3. the magnitude of the impulse exerted by \(B\) on the ground,
4. the time between the instant when \(B\) strikes the ground and the instant when \(A\) reaches its highest point.

2. A particle \(P\) is moving with constant velocity ( \(2 \mathbf { i } - 3 \mathbf { j }\) ) \(\mathrm { m } \mathrm { s } ^ { - 1 }\).

1. Find the speed of \(P\). The particle \(P\) passes through the point \(A\) and 4 seconds later passes through the point with position vector ( \(\mathbf { i } - 4 \mathbf { j }\) ) m.
2. Find the position vector of \(A\).

8. [In this question \(\mathbf { i }\) and \(\mathbf { j }\) are horizontal unit vectors directed due east and due north respectively and position vectors are given relative to a fixed origin.] A ship \(A\) moves with constant velocity \(( 3 \mathbf { i } - 10 \mathbf { j } ) \mathrm { kmh } ^ { - 1 }\) At time \(t\) hours, the position vector of \(A\) is \(\mathbf { r } \mathrm { km }\).
At time \(t = 0 , A\) is at the point with position vector \(( 13 \mathbf { i } + 5 \mathbf { j } ) \mathrm { km }\).

1. Find \(\mathbf { r }\) in terms of \(t\). Another ship \(B\) moves with constant velocity \(( 15 \mathbf { i } + 14 \mathbf { j } ) \mathrm { km } \mathrm { h } ^ { - 1 }\) At time \(t = 0 , B\) is at the point with position vector \(( 3 \mathbf { i } - 5 \mathbf { j } ) \mathrm { km }\).
2. Show that, at time \(t\) hours, $$\overrightarrow { A B } = [ ( 12 t - 10 ) \mathbf { i } + ( 24 t - 10 ) \mathbf { j } ] \mathrm { km }$$ Given that the two ships do not change course,
3. find the shortest distance between the two ships,
4. find the bearing of ship \(B\) from ship \(A\) when the ships are closest.
   
   \includegraphics[max width=\textwidth, alt={}]{f1bdc84b-c8a1-4e7c-a2ba-48b40c6a6d36-28_2820_1967_102_100}

1. \hspace{0pt} [In this question \(\mathbf { i }\) and \(\mathbf { j }\) are horizontal unit vectors directed due east and due north respectively and position vectors are given relative to a fixed origin \(O\).]

At midnight, a ship \(S\) is at the point with position vector ( \(19 \mathbf { i } + 22 \mathbf { j }\) )km
The ship travels with constant velocity \(( 12 \mathbf { i } - 16 \mathbf { j } ) \mathrm { km } \mathrm { h } ^ { - 1 }\)

1. Find the speed of \(S\). At time \(t\) hours after midnight, the position vector of \(S\) is \(\mathbf { s } \mathrm { km }\).
2. Find an expression for \(\mathbf { s }\) in terms of \(\mathbf { i } , \mathbf { j }\) and \(t\). A lighthouse stands on a small rocky island. The lighthouse is modelled as being at the point with position vector \(( 26 \mathbf { i } + 15 \mathbf { j } ) \mathrm { km }\). It is not safe for ships to be within 1.3 km of the lighthouse.
3. 1. Find the value of \(t\) when \(S\) is closest to the lighthouse.
   2. Hence determine whether it is safe for \(S\) to continue its course.

5. [In this question \(\mathbf { i }\) and \(\mathbf { j }\) are perpendicular horizontal unit vectors and position vectors are given relative to a fixed origin \(O\).] A particle \(P\) is moving in a straight line with constant velocity. At 9 am, the position vector of \(P\) is \(( 7 \mathbf { i } + 5 \mathbf { j } ) \mathrm { km }\) and at 9.20 am , the position vector of \(P\) is \(6 \mathbf { i } \mathrm {~km}\). At time \(t\) hours after 9 am , the position vector of \(P\) is \(\mathbf { r } _ { P } \mathrm {~km}\).

1. Find, in \(\mathrm { kmh } ^ { - 1 }\), the speed of \(P\).
2. Show that \(\mathbf { r } _ { P } = ( 7 - 3 t ) \mathbf { i } + ( 5 - 15 t ) \mathbf { j }\).
3. Find the value of \(t\) when \(\mathbf { r } _ { P }\) is parallel to \(16 \mathbf { i } + 5 \mathbf { j }\). The position vector of another particle \(Q\), at time \(t\) hours after 9 am , is \(\mathbf { r } _ { Q } \mathrm {~km}\), where \(\mathbf { r } _ { Q } = ( 5 + 2 t ) \mathbf { i } + ( - 3 + 5 t ) \mathbf { j }\)
4. Show that \(P\) and \(Q\) will collide and find the position vector of the point of collision.

8. [In this question, \(\mathbf { i }\) and \(\mathbf { j }\) are horizontal unit vectors directed due east and due north respectively and position vectors are given relative to a fixed origin \(O\).] Two boats, \(P\) and \(Q\), are moving with constant velocities.
The velocity of \(P\) is \(15 \mathbf { i } \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and the velocity of \(Q\) is \(( 20 \mathbf { i } - 20 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\)

1. Find the direction in which \(Q\) is travelling, giving your answer as a bearing. The boats are modelled as particles.
   At time \(t = 0 , P\) is at the origin \(O\) and \(Q\) is at the point with position vector \(200 \mathbf { j } \mathrm {~m}\). At time \(t\) seconds, the position vector of \(P\) is \(\mathbf { p m }\) and the position vector of \(Q\) is \(\mathbf { q m }\).
2. Show that $$\overrightarrow { P Q } = [ 5 t \mathbf { i } + ( 200 - 20 t ) \mathbf { j } ] \mathrm { m }$$
3. Find the bearing of \(P\) from \(Q\) when \(t = 10\)
4. Find the distance between \(P\) and \(Q\) when \(Q\) is north east of \(P\)
5. Find the times when \(P\) and \(Q\) are 200 m apart.

8. \begin{figure}[h] \end{figure} A square floor space \(A B C D\), with centre \(O\), is modelled as a flat horizontal surface measuring 50 m by 50 m , as shown in Figure 5 .
The horizontal unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are in the direction of \(\overrightarrow { A B }\) and \(\overrightarrow { A D }\) respectively.
All position vectors are given relative to \(O\).
A small robot \(R\) is programmed to travel across the floor at a constant velocity.

• At time \(t = 0 , R\) is at the point with position vector ( \(- 2 \mathbf { i } + \mathbf { j }\) ) m
• At time \(t = 11 \mathrm {~s} , R\) is at the point with position vector \(( 9 \mathbf { i } + 23 \mathbf { j } ) \mathrm { m }\)
• At time \(t\) seconds, the position vector of \(R\) is \(\mathbf { r }\) metres
  1. Find, in terms of \(t\), i and j, an expression for \(\mathbf { r }\)

A second robot \(S\) is at the point \(C\).

$$\overrightarrow { S R } = [ ( 2 t - 27 ) \mathbf { i } + ( 3 t - 24 ) \mathbf { j } ] \mathbf { m }$$

Find the time when the distance between \(R\) and \(S\) is a minimum.

1. \hspace{0pt} [In this question, the horizontal unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are directed due east and due north respectively and position vectors are given relative to a fixed origin \(O\).]

Two speedboats, \(A\) and \(B\), are each moving with constant velocity.

• the velocity of \(A\) is \(40 \mathrm { kmh } ^ { - 1 }\) due east
• the velocity of \(B\) is \(20 \mathrm { kmh } ^ { - 1 }\) on a bearing of angle \(\alpha \left( 0 ^ { \circ } < \alpha < 90 ^ { \circ } \right)\), where \(\tan \alpha = \frac { 4 } { 3 }\) The boats are modelled as particles.
  1. Find, in terms of \(\mathbf { i }\) and \(\mathbf { j }\), the velocity of \(B\) in \(\mathrm { km } \mathrm { h } ^ { - 1 }\)

At noon

• the position vector of \(A\) is \(20 \mathbf { j } \mathrm {~km}\)
• the position vector of \(B\) is \(( 10 \mathbf { i } + 5 \mathbf { j } ) \mathrm { km }\)

At time \(t\) hours after noon

• the position vector of \(A\) is \(\mathbf { r k m }\), where \(\mathbf { r } = 20 \mathbf { j } + 40 t \mathbf { i }\)
• the position vector of \(B\) is \(\mathbf { s }\) km
• Find an expression for \(\mathbf { s }\) in terms of \(t , \mathbf { i }\) and \(\mathbf { j }\).
• Show that at time \(t\) hours after noon,

$$\overrightarrow { A B } = [ ( 10 - 24 t ) \mathbf { i } + ( 12 t - 15 ) \mathbf { j } ] \mathrm { km }$$

Show that the boats will never collide.

Find the distance between the boats when the bearing of \(B\) from \(A\) is \(225 ^ { \circ }\)

1. \hspace{0pt} [In this question \(\mathbf { i }\) and \(\mathbf { j }\) are horizontal unit vectors due east and due north respectively and position vectors are given relative to a fixed origin \(O\) ]

A particle \(P\) is moving with velocity \(( \mathbf { i } - 2 \mathbf { j } ) \mathrm { km } \mathrm { h } ^ { - 1 }\). At time \(t = 0\) hours, the position vector of \(P\) is \(( - 5 \mathbf { i } + 9 \mathbf { j } ) \mathrm { km }\). At time \(t\) hours, the position vector of \(P\) is \(\mathbf { p } \mathrm { km }\).

1. Find an expression for \(\mathbf { p }\) in terms of \(t\). The point \(A\) has position vector ( \(3 \mathbf { i } + 2 \mathbf { j }\) ) km.
2. Find the position vector of \(P\) when \(P\) is due west of \(A\). Another particle \(Q\) is moving with velocity \([ ( 2 b - 1 ) \mathbf { i } + ( 5 - 2 b ) \mathbf { j } ] \mathrm { km } \mathrm { h } ^ { - 1 }\) where \(b\) is a constant. Given that the particles are moving along parallel lines,
3. find the value of \(b\).

6. The point \(A\) on a horizontal playground has position vector \(( 3 \mathbf { i } - 2 \mathbf { j } ) \mathrm { m }\). At time \(t = 0\), a girl kicks a ball from \(A\). The ball moves horizontally along the playground with constant velocity \(( 4 \mathbf { i } + 5 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\). Modelling the ball as a particle, find

1. the speed of the ball,
2. the position vector of the ball at time \(t\) seconds. The point \(B\) on the playground has position vector \(( \mathbf { i } + 6 \mathbf { j } ) \mathrm { m }\). At time \(t = T\) seconds, the ball is due east of \(B\).
3. Find the value of \(T\). A boy is running due east with constant speed \(\nu \mathrm { ms } ^ { - 1 }\). At the instant when the girl kicks the ball from \(A\), the boy is at \(B\). Given that the boy intercepts the ball,
4. find the value of \(v\). \includegraphics[max width=\textwidth, alt={}, center]{5f2d38d9-b719-4205-8cb0-caa959afc46f-23_68_47_2617_1886}

8. [In this question \(\mathbf { i }\) and \(\mathbf { j }\) are horizontal unit vectors directed due east and due north respectively and position vectors are given relative to a fixed origin.] At 7 am a ship leaves a port and moves with constant velocity. The position vector of the port is \(( - 2 \mathbf { i } + 9 \mathbf { j } ) \mathrm { km }\). At 7.36 am the ship is at the point with position vector \(( 4 \mathbf { i } + 6 \mathbf { j } ) \mathrm { km }\).

1. Show that the velocity of the ship is \(( 10 \mathbf { i } - 5 \mathbf { j } ) \mathrm { km } \mathrm { h } ^ { - 1 }\)
2. Find the position vector of the ship \(t\) hours after leaving port. At 8.48 am a passenger on the ship notices that a lighthouse is due east of the ship. At 9 am the same passenger notices that the lighthouse is now north east of the ship.
3. Find the position vector of the lighthouse.
4. Find the position vector of the ship when it is due south of the lighthouse.
   
   \includegraphics[max width=\textwidth, alt={}]{151d9232-5a78-4bc1-a57e-6c9cae80e473-32_2258_53_308_1980}

1. \hspace{0pt} [In this question \(\mathbf { i }\) and \(\mathbf { j }\) are horizontal unit vectors directed due east and due north respectively and position vectors are given relative to a fixed origin \(O\).]

At 12:00, a ship \(P\) sets sail from a harbour with position vector \(( 15 \mathbf { i } + 36 \mathbf { j } ) \mathrm { km }\). At 12:30, \(P\) is at the point with position vector \(( 20 \mathbf { i } + 34 \mathbf { j } ) \mathrm { km }\). Given that \(P\) moves with constant velocity,

1. show that the velocity of \(P\) is \(( 10 \mathbf { i } - 4 \mathbf { j } ) \mathrm { kmh } ^ { - 1 }\) At time \(t\) hours after 12:00, the position vector of \(P\) is \(\mathbf { p } \mathrm { km }\).
2. Find an expression for \(\mathbf { p }\) in terms of \(\mathbf { i } , \mathbf { j }\) and \(t\). A second ship \(Q\) is also travelling at a constant velocity.
   At time \(t\) hours after 12:00, the position vector of \(Q\) is given by \(\mathbf { q } \mathrm { km }\), where $$\mathbf { q } = ( 42 - 8 t ) \mathbf { i } + ( 9 + 14 t ) \mathbf { j }$$ Ships \(P\) and \(Q\) are modelled as particles.
   If both ships maintained their course,
3. 1. verify that they would collide at 13:30
   2. find the position vector of the point at which the collision would occur. At 12:30 \(Q\) changes speed and direction to avoid the collision.
      Ship \(Q\) now travels due north with a constant speed of \(15 \mathrm { kmh } ^ { - 1 }\) Ship \(P\) maintains the same constant velocity throughout.
4. Find the exact distance between \(P\) and \(Q\) at 14:30

4 A projectile P travels in a vertical plane over level ground. Its position vector \(\mathbf { r }\) at time \(t\) seconds after projection is modelled by $$\mathbf { r } = \binom { x } { y } = \binom { 0 } { 5 } + \binom { 30 } { 40 } t - \binom { 0 } { 5 } t ^ { 2 } ,$$ where distances are in metres and the origin is a point on the level ground.

1. Write down
   (A) the height from which P is projected,
   (B) the value of \(g\) in this model.
2. Find the displacement of P from \(t = 3\) to \(t = 5\).
3. Show that the equation of the trajectory is $$y = 5 + \frac { 4 } { 3 } x - \frac { x ^ { 2 } } { 180 } .$$

4 \includegraphics[max width=\textwidth, alt={}, center]{798da17d-0af5-4aa6-b731-564642dc28d5-2_428_572_861_760} As shown in the diagram the points \(A\) and \(B\) have position vectors \(\mathbf { a }\) and \(\mathbf { b }\) with respect to the origin \(O\).

1. Make a sketch of the diagram, and mark the points \(C , D\) and \(E\) such that \(\overrightarrow { O C } = 2 \mathbf { a } , \overrightarrow { O D } = 2 \mathbf { a } + \mathbf { b }\) and \(\overrightarrow { O E } = \frac { 1 } { 3 } \overrightarrow { O D }\).
2. By expressing suitable vectors in terms of \(\mathbf { a }\) and \(\mathbf { b }\), prove that \(E\) lies on the line joining \(A\) and \(B\).

4 The points \(\mathrm { A } , \mathrm { B }\) and C are given by the position vectors \(\mathbf { a } = \binom { - 2 } { 1 } , \mathbf { b } = \binom { 0 } { 5 }\) and \(\mathbf { c } = \binom { 4 } { 3 }\). M is the midpoint of AC .

1. Find the position vector of M .
2. Find the vector \(\overrightarrow { B C }\).
3. Find the position vector of the point D such that \(\overrightarrow { \mathrm { BC } } = \overrightarrow { \mathrm { AD } }\).
4. Show that D lies on BM .