Bearing and speed from velocity vector

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. [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 ships, \(P\) and \(Q\), are moving with constant velocities.
The velocity of \(P\) is \(( 9 \mathbf { i } - 2 \mathbf { j } ) \mathrm { km } \mathrm { h } ^ { - 1 }\) and the velocity of \(Q\) is \(( 4 \mathbf { i } + 8 \mathbf { j } ) \mathrm { km } \mathrm { h } ^ { - 1 }\)

1. Find the direction of motion of \(P\), giving your answer as a bearing to the nearest degree. When \(t = 0\), the position vector of \(P\) is \(( 9 \mathbf { i } + 10 \mathbf { j } ) \mathrm { km }\) and the position vector of \(Q\) is \(( \mathbf { i } + 4 \mathbf { j } ) \mathrm { km }\). At time \(t\) hours, the position vectors of \(P\) and \(Q\) are \(\mathbf { p } \mathrm { km }\) and \(\mathbf { q } \mathrm { km }\) respectively.
2. Find an expression for
   1. \(\mathbf { p }\) in terms of \(t\),
   2. \(\mathbf { q }\) in terms of \(t\).
3. Hence show that, at time \(t\) hours, $$\overrightarrow { Q P } = ( 8 + 5 t ) \mathbf { i } + ( 6 - 10 t ) \mathbf { j }$$
4. Find the values of \(t\) when the ships are 10 km apart.

6 The velocity of a model boat, \(\mathbf { v } \mathrm { m } \mathrm { s } ^ { - 1 }\), is given by $$\mathbf { v } = \binom { - 5 } { 10 } + t \binom { 6 } { - 8 }$$ where \(t\) is the time in seconds and the vectors \(\binom { 1 } { 0 }\) and \(\binom { 0 } { 1 }\) are east and north respectively.

1. Show that when \(t = 2.5\) the boat is travelling south-east (i.e. on a bearing of \(135 ^ { \circ }\) ). Calculate its speed at this time. The boat is at a point O when \(t = 0\).
2. Calculate the bearing of the boat from O when \(t = 2.5\).

1. \hspace{0pt} [In this question the unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are due east and due north respectively.]

A coastguard station \(O\) monitors the movements of a small boat.
At 10:00 the boat is at the point \(( 4 \mathbf { i } - 2 \mathbf { j } ) \mathrm { km }\) relative to \(O\).
At 12:45 the boat is at the point \(( - 3 \mathbf { i } - 5 \mathbf { j } ) \mathrm { km }\) relative to \(O\).
The motion of the boat is modelled as that of a particle moving in a straight line at constant speed.

1. Calculate the bearing on which the boat is moving, giving your answer in degrees to one decimal place.
2. Calculate the speed of the boat, giving your answer in \(\mathrm { kmh } ^ { - 1 }\)

1. \hspace{0pt} [In this question \(\mathbf { i }\) and \(\mathbf { j }\) are unit vectors due east and due north respectively.]

A ship \(P\) is moving with velocity ( \(5 \mathbf { i } - 4 \mathbf { j }\) ) \(\mathrm { km } \mathrm { h } ^ { - 1 }\) and a ship \(Q\) is moving with velocity \(( 3 \mathbf { i } + 7 \mathbf { j } ) \mathrm { km } \mathrm { h } ^ { - 1 }\). Find the direction that ship \(Q\) appears to be moving in, to an observer on ship \(P\), giving your answer as a bearing.

1. \hspace{0pt} [In this question \(\mathbf { i }\) and \(\mathbf { j }\) are horizontal unit vectors due east and due north respectively.]

Boat \(A\) is moving with velocity ( \(3 \mathbf { i } + 4 \mathbf { j } ) \mathrm { km } \mathrm { h } ^ { - 1 }\) and boat \(B\) is moving with velocity \(( 6 \mathbf { i } - 5 \mathbf { j } ) \mathrm { km } \mathrm { h } ^ { - 1 }\). Find

1. the magnitude of the velocity of \(A\) relative to \(B\),
2. the direction of the velocity of \(A\) relative to \(B\), giving your answer as a bearing.

18 In this question \(\mathbf { i }\) and \(\mathbf { j }\) are perpendicular unit vectors representing due east and due north respectively. A particle, \(T\), is moving on a plane at a constant speed.
The path followed by \(T\) makes the exact shape of a triangle \(A B C\). \(T\) moves around \(A B C\) in an anticlockwise direction as shown in the diagram below. \includegraphics[max width=\textwidth, alt={}, center]{de8a7d38-a665-4feb-854e-ac83f413d133-28_447_366_671_925} On its journey from \(A\) to \(B\) the velocity vector of \(T\) is \(( 3 \mathbf { i } + \sqrt { 3 } \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\) 18

1. Find the speed of \(T\) as it moves from \(A\) to \(B\) 18
2. On its journey from \(B\) to \(C\) the velocity vector of \(T\) is \(( - 3 \mathbf { i } + \sqrt { 3 } \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\) Show that the acute angle \(A B C = 60 ^ { \circ }\) 18
3. It is given that \(A B C\) is an equilateral triangle. \(T\) returns to its initial position after 9 seconds.
   Vertex \(B\) lies at position vector \(\left[ \begin{array} { l } 1 \\ 0 \end{array} \right]\) metres with respect to a fixed origin \(O\) Find the position vector of \(C\)

[In this question \(\mathbf{i}\) and \(\mathbf{j}\) are unit vectors directed due east and due north respectively.] A particle \(P\) is moving with constant velocity \((-6\mathbf{i} + 2\mathbf{j})\) m s\(^{-1}\). At time \(t = 0\), \(P\) passes through the point with position vector \((21\mathbf{i} + 5\mathbf{j})\) m, relative to a fixed origin \(O\).

1. Find the direction of motion of \(P\), giving your answer as a bearing to the nearest degree. [3]
2. Write down the position vector of \(P\) at time \(t\) seconds. [1]
3. Find the time at which \(P\) is north-west of \(O\). [3]

[In this question, the horizontal unit vectors \(\mathbf{i}\) and \(\mathbf{j}\) are directed due east and due north respectively.] The velocity, \(\mathbf{v} \text{ m s}^{-1}\), of a particle \(P\) at time \(t\) seconds is given by $$\mathbf{v} = (1 - 2t)\mathbf{i} + (3t - 3)\mathbf{j}$$

1. Find the speed of \(P\) when \(t = 0\) [3]
2. Find the bearing on which \(P\) is moving when \(t = 2\) [2]
3. Find the value of \(t\) when \(P\) is moving
   1. parallel to \(\mathbf{j}\),
   2. parallel to \((-\mathbf{i} - 3\mathbf{j})\). [6]

A ship \(S\) is moving with constant velocity \((4\mathbf{i} - 7\mathbf{j})\text{ms}^{-1}\), where \(\mathbf{i}\) and \(\mathbf{j}\) are unit vectors due east and due north respectively. Find the speed and direction of \(S\), giving the direction as a three-figure bearing, correct to the nearest degree. [4]