When is one object due north/east/west/south of another

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 \(Q\).

2. [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 \(O\).] At time \(t = 0\), a bird \(A\) leaves its nest, that is located at the point with position vector \(( 20 \mathbf { i } - 17 \mathbf { j } ) \mathrm { m }\), and flies with constant velocity \(( - 6 \mathbf { i } + 7 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\). At the same time a second bird \(B\) leaves its nest which is located at the point with position vector \(( - 8 \mathbf { i } + 9 \mathbf { j } ) \mathrm { m }\) and flies with constant velocity ( \(p \mathbf { i } + 2 p \mathbf { j }\) ) \(\mathrm { m } \mathrm { s } ^ { - 1 }\), where \(p\) is a constant. At time \(t = 4 \mathrm {~s}\), bird \(B\) is south west of bird \(A\).

1. Find the direction of motion of \(A\), giving your answer as a bearing to the nearest degree.
2. Find the speed of \(B\).

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.

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}

1. A \(\operatorname { ship } A\) is moving at a constant speed of \(8 \mathrm {~km} \mathrm {~h} \mathrm {~h} ^ { - 1 }\) on a bearing of \(150 ^ { \circ }\). At noon a second ship \(B\) is 6 km from \(A\), on a bearing of \(210 ^ { \circ }\). Ship \(B\) is moving due east at a constant speed. At a later time, \(B\) is \(2 \sqrt { 3 } \mathrm {~km}\) due south of \(A\).

Find

1. the time at which \(B\) will be due east of \(A\),
2. the distance between the ships at that time.

Two ships \(P\) and \(Q\) are moving along straight lines with constant velocities. Initially \(P\) is at a point \(O\) and the position vector of \(Q\) relative to \(O\) is \((6\mathbf{i} + 12\mathbf{j})\) km, where \(\mathbf{i}\) and \(\mathbf{j}\) are unit vectors directed due east and due north respectively. The ship \(P\) is moving with velocity \(10\mathbf{j}\) km h\(^{-1}\) and \(Q\) is moving with velocity \((-8\mathbf{i} + 6\mathbf{j})\) km h\(^{-1}\). At time \(t\) hours the position vectors of \(P\) and \(Q\) relative to \(O\) are \(\mathbf{p}\) km and \(\mathbf{q}\) km respectively.

1. Find \(\mathbf{p}\) and \(\mathbf{q}\) in terms of \(t\). [3]
2. Calculate the distance of \(Q\) from \(P\) when \(t = 3\). [3]
3. Calculate the value of \(t\) when \(Q\) is due north of \(P\). [2]

[In this question the vectors \(\mathbf{i}\) and \(\mathbf{j}\) are horizontal unit vectors in the direction due east and due north respectively.] Two boats \(A\) and \(B\) are moving with constant velocities. Boat \(A\) moves with velocity \(9\mathbf{j}\) km h\(^{-1}\). Boat \(B\) moves with velocity \((3\mathbf{i} + 5\mathbf{j})\) km h\(^{-1}\).

1. Find the bearing on which \(B\) is moving. [2]

At noon, \(A\) is at point \(O\), and \(B\) is 10 km due west of \(O\). At time \(t\) hours after noon, the position vectors of \(A\) and \(B\) relative to \(O\) are \(\mathbf{a}\) km and \(\mathbf{b}\) km respectively.

1. Find expressions for \(\mathbf{a}\) and \(\mathbf{b}\) in terms of \(t\), giving your answer in the form \(p\mathbf{i} + q\mathbf{j}\). [3]
2. Find the time when \(B\) is due south of \(A\). [2]

At time \(t\) hours after noon, the distance between \(A\) and \(B\) is \(d\) km. By finding an expression for \(\overrightarrow{AB}\),

1. show that \(d^2 = 25t^2 - 60t + 100\). [4]

At noon, the boats are 10 km apart.

1. Find the time after noon at which the boats are again 10 km apart. [3]

[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 sets sail at 9 am from a port \(P\) and moves with constant velocity. The position vector of \(P\) is \((4\mathbf{i} - 8\mathbf{j})\) km. At 9.30 am the ship is at the point with position vector \((\mathbf{i} - 4\mathbf{j})\) km.

1. Find the speed of the ship in km h\(^{-1}\). [4]
2. Show that the position vector \(\mathbf{r}\) km of the ship, \(t\) hours after 9 am, is given by \(\mathbf{r} = (4 - 6t)\mathbf{i} + (8t - 8)\mathbf{j}\). [2]

At 10 am, a passenger on the ship observes that a lighthouse \(L\) is due west of the ship. At 10.30 am, the passenger observes that \(L\) is now south-west of the ship.

1. Find the position vector of \(L\). [5]

[In this question \(\mathbf{i}\) and \(\mathbf{j}\) are unit vectors due east and due north respectively. Position vectors are given relative to a fixed origin \(O\).] Two ships \(P\) and \(Q\) are moving with constant velocities. Ship \(P\) moves with velocity \((2\mathbf{i} - 3\mathbf{j})\) km h\(^{-1}\) and ship \(Q\) moves with velocity \((3\mathbf{i} + 4\mathbf{j})\) km h\(^{-1}\).

1. Find, to the nearest degree, the bearing on which \(Q\) is moving. [2]

At 2 pm, ship \(P\) is at the point with position vector \((\mathbf{i} + \mathbf{j})\) km and ship \(Q\) is at the point with position vector \((-2\mathbf{j})\) km. At time \(t\) hours after 2 pm, the position vector of \(P\) is \(\mathbf{p}\) km and the position vector of \(Q\) is \(\mathbf{q}\) km.

1. Write down expressions, in terms of \(t\), for
   1. \(\mathbf{p}\),
   2. \(\mathbf{q}\),
   3. \(\overrightarrow{PQ}\). [5]
2. Find the time when
   1. \(Q\) is due north of \(P\),
   2. \(Q\) is north-west of \(P\). [4]