1.10a Vectors in 2D: i,j notation and column vectors

8. With respect to a fixed origin \(O\), the line \(l _ { 1 }\) is given by the equation $$\mathbf { r } = \left( \begin{array} { r } 8 \\ 1 \\ - 3 \end{array} \right) + \mu \left( \begin{array} { r } - 5 \\ 4 \\ 3 \end{array} \right)$$ where \(\mu\) is a scalar parameter.
The point \(A\) lies on \(l _ { 1 }\) where \(\mu = 1\)

1. Find the coordinates of \(A\). The point \(P\) has position vector \(\left( \begin{array} { l } 1 \\ 5 \\ 2 \end{array} \right)\).
   The line \(l _ { 2 }\) passes through the point \(P\) and is parallel to the line \(l _ { 1 }\)
2. Write down a vector equation for the line \(l _ { 2 }\)
3. Find the exact value of the distance \(A P\). Give your answer in the form \(k \sqrt { 2 }\), where \(k\) is a constant to be determined. The acute angle between \(A P\) and \(l _ { 2 }\) is \(\theta\).
4. Find the value of \(\cos \theta\) A point \(E\) lies on the line \(l _ { 2 }\) Given that \(A P = P E\),
5. find the area of triangle \(A P E\),
6. find the coordinates of the two possible positions of \(E\).

6. With respect to a fixed origin \(O\), the lines \(l _ { 1 }\) and \(l _ { 2 }\) are given by the equations $$l _ { 1 } : \mathbf { r } = \left( \begin{array} { r } 4 \\ 28 \\ 4 \end{array} \right) + \lambda \left( \begin{array} { r } - 1 \\ - 5 \\ 1 \end{array} \right) , \quad l _ { 2 } : \mathbf { r } = \left( \begin{array} { l } 5 \\ 3 \\ 1 \end{array} \right) + \mu \left( \begin{array} { r } 3 \\ 0 \\ - 4 \end{array} \right)$$ where \(\lambda\) and \(\mu\) are scalar parameters. The lines \(l _ { 1 }\) and \(l _ { 2 }\) intersect at the point \(X\).

1. Find the coordinates of the point \(X\).
2. Find the size of the acute angle between \(l _ { 1 }\) and \(l _ { 2 }\), giving your answer in degrees to 2 decimal places. The point \(A\) lies on \(l _ { 1 }\) and has position vector \(\left( \begin{array} { r } 2 \\ 18 \\ 6 \end{array} \right)\)
3. Find the distance \(A X\), giving your answer as a surd in its simplest form. The point \(Y\) lies on \(l _ { 2 }\). Given that the vector \(\overrightarrow { Y A }\) is perpendicular to the line \(l _ { 1 }\)
4. find the distance \(Y A\), giving your answer to one decimal place. The point \(B\) lies on \(l _ { 1 }\) where \(| \overrightarrow { A X } | = 2 | \overrightarrow { A B } |\).
5. Find the two possible position vectors of \(B\).

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}

5. A particle \(P\) moves with constant acceleration \(( 2 \mathbf { i } - 3 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 2 }\). At time \(t\) seconds, its velocity is \(\mathbf { v } \mathrm { m } \mathrm { s } ^ { - 1 }\). When \(t = 0 , \mathbf { v } = - 2 \mathbf { i } + 7 \mathbf { j }\).

1. Find the value of \(t\) when \(P\) is moving parallel to the vector \(\mathbf { i }\).
2. Find the speed of \(P\) when \(t = 3\).
3. Find the angle between the vector \(\mathbf { j }\) and the direction of motion of \(P\) when \(t = 3\).

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 \(B\).

7. Two forces \(\mathbf { F } _ { 1 }\) and \(\mathbf { F } _ { 2 }\) act on a particle \(P\). The force \(\mathbf { F } _ { 1 }\) is given by \(\mathbf { F } _ { 1 } = ( - \mathbf { i } + 2 \mathbf { j } ) \mathrm { N }\) and \(\mathbf { F } _ { 2 }\) acts in the direction of the vector \(( \mathbf { i } + \mathbf { j } )\).
Given that the resultant of \(\mathbf { F } _ { 1 }\) and \(\mathbf { F } _ { 2 }\) acts in the direction of the vector ( \(\mathbf { i } + 3 \mathbf { j }\) ),

1. find \(\mathbf { F } _ { 2 }\) (7) The acceleration of \(P\) is \(( 3 \mathbf { i } + 9 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 2 }\). At time \(t = 0\), the velocity of \(P\) is \(( 3 \mathbf { i } - 22 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\)
2. Find the speed of \(P\) when \(t = 3\) seconds.
   

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. [In this question \(\mathbf { i }\) and \(\mathbf { j }\) are horizontal unit vectors due east and due north respectively] Two forces \(\mathbf { F } _ { 1 }\) and \(\mathbf { F } _ { 2 }\) act on a particle \(P\) of mass 0.5 kg . \(\mathbf { F } _ { 1 } = ( 4 \mathbf { i } - 6 \mathbf { j } ) \mathrm { N }\) and \(\mathbf { F } _ { 2 } = ( p \mathbf { i } + q \mathbf { j } ) \mathrm { N }\).
Given that the resultant force of \(\mathbf { F } _ { 1 }\) and \(\mathbf { F } _ { 2 }\) is in the same direction as \(- 2 \mathbf { i } - \mathbf { j }\),

1. show that \(p - 2 q = - 16\) Given that \(q = 3\)
2. find the magnitude of the acceleration of \(P\),
3. find the direction of the acceleration of \(P\), giving your answer as a bearing to the nearest degree. XXXXXXXXXXIXITEINTIIS AREA XX女X女X女X女X DO NOT WIRIE IN THS AREA.

1. \begin{figure}[h] \end{figure} A particle \(P\) of weight 6 N is attached to one end of a light inextensible string. The other end of the string is attached to a fixed point \(O\). A horizontal force of magnitude \(F\) newtons is applied to \(P\). The particle \(P\) is in equilibrium under gravity with the string making an angle of \(30 ^ { \circ }\) with the vertical, as shown in Fig. 1. Find, to 3 significant figures,

1. the tension in the string,
2. the value of \(F\).

2. A particle \(P\) of mass 1.5 kg is moving under the action of a constant force ( \(3 \mathbf { i } - 7.5 \mathbf { j }\) ) N. Initially \(P\) has velocity \(( 2 \mathbf { i } + 3 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\). Find

1. the magnitude of the acceleration of \(P\),
2. the velocity of \(P\), in terms of \(\mathbf { i }\) and \(\mathbf { j }\), when \(P\) has been moving for 4 seconds.

5. A t time \(t\) seconds ( \(t \geqslant 0\) ), a particle \(P\) has velocity \(\mathbf { v m ~ s } ^ { - 1 }\), where $$\mathbf { v } = \left( 3 t ^ { 2 } - 4 \right) \mathbf { i } + ( 2 t - 4 ) \mathbf { j }$$ When \(t = 0 , P\) is at the fixed point \(O\).

1. Find the acceleration of \(P\) at the instant when \(t = 0\)
2. Find the exact speed of \(P\) at the instant when \(P\) is moving in the direction of the vector \(( 11 \mathbf { i } + \mathbf { j } )\) for the second time.
3. Show that \(P\) never returns to \(O\). \includegraphics[max width=\textwidth, alt={}, center]{c16c17b6-2c24-4939-b3b5-63cd63646b76-14_2658_1938_107_123} \includegraphics[max width=\textwidth, alt={}, center]{c16c17b6-2c24-4939-b3b5-63cd63646b76-15_149_140_2604_1818}

5. At time \(t\) seconds, \(t \geqslant 0\), a particle \(P\) has velocity \(\mathbf { v } \mathrm { ms } ^ { - 1 }\), where $$\mathbf { v } = \left( 5 t ^ { 2 } - 12 t + 15 \right) \mathbf { i } + \left( t ^ { 2 } + 8 t - 10 \right) \mathbf { j }$$ When \(t = 0 , P\) is at the origin \(O\).
At time \(T\) seconds, \(P\) is moving in the direction of \(( \mathbf { i } + \mathbf { j } )\).

1. Find the value of \(T\). When \(t = 3 , P\) is at the point \(A\).
2. Find the magnitude of the acceleration of \(P\) as it passes through \(A\).
3. Find the position vector of \(A\).

3. A particle \(P\) of mass 0.25 kg is moving on a smooth horizontal surface under the action of a single force, \(\mathbf { F }\) newtons. At time \(t\) seconds \(( t \geqslant 0 )\), the velocity \(\mathbf { v } \mathrm { m } \mathrm { s } ^ { - 1 }\) of \(P\) is given by $$\mathbf { v } = ( 6 \sin 3 t ) \mathbf { i } + ( 1 + 2 \cos t ) \mathbf { j }$$

1. Find \(\mathbf { F }\) in terms of \(t\). At time \(t = 0\), the position vector of \(P\) relative to a fixed point \(O\) is \(( 4 \mathbf { i } - \sqrt { 3 } \mathbf { j } ) \mathrm { m }\).
2. Find the position vector of \(P\) relative to \(O\) when \(P\) is first moving parallel to the vector \(\mathbf { i }\).

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

A particle \(P\) of mass 2 kg moves under the action of two forces, \(( 2 \mathbf { i } + 3 \mathbf { j } ) \mathrm { N }\) and \(( 4 \mathbf { i } - 5 \mathbf { j } ) \mathrm { N }\).

1. Find the magnitude of the acceleration of \(P\). At time \(t = 0 , P\) has velocity ( \(- u \mathbf { i } + u \mathbf { j }\) ) \(\mathrm { m } \mathrm { s } ^ { - 1 }\), where \(u\) is a positive constant. At time \(t = T\) seconds, \(P\) has velocity \(( 10 \mathbf { i } + 2 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\).
2. Find
   1. the value of \(T\),
   2. the value of \(u\).

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 \(B\).

6. Two girls, Agatha and Brionie, are roller skating inside a large empty building. The girls are modelled as particles. At time \(t = 0\), Agatha is at the point with position vector \(( 11 \mathbf { i } + 11 \mathbf { j } ) \mathrm { m }\) and Brionie is at the point with position vector \(( 7 \mathbf { i } + 16 \mathbf { j } ) \mathrm { m }\). The position vectors are given relative to the door, \(O\), and \(\mathbf { i }\) and \(\mathbf { j }\) are horizontal perpendicular unit vectors. Agatha skates with constant velocity ( \(3 \mathbf { i } - \mathbf { j }\) ) \(\mathrm { m } \mathrm { s } ^ { - 1 }\) Brionie skates with constant velocity ( \(4 \mathbf { i } - 2 \mathbf { j }\) ) \(\mathrm { m } \mathrm { s } ^ { - 1 }\)

1. Find the position vector of Agatha at time \(t\) seconds. At time \(t = 6\) seconds, Agatha passes through the point \(P\).
2. Show that Brionie also passes through \(P\) and find the value of \(t\) when this occurs. At time \(t\) seconds, Agatha is at the point \(A\) and Brionie is at the point \(B\).
3. Show that \(\overrightarrow { A B } = [ ( t - 4 ) \mathbf { i } + ( 5 - t ) \mathbf { j } ] \mathrm { m }\)
4. Find the distance between the two girls when they are closest together. \includegraphics[max width=\textwidth, alt={}, center]{ca445c1e-078c-4a57-94df-de90f30f8efd-13_2255_50_314_34}
   

   VIXV SIHIANI III IM IONOO

   VIAV SIHI NI JYHAM ION OO

   VI4V SIHI NI JLIYM ION OO

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 \(225 ^ { \circ }\)

2. [In this question \(\mathbf { i }\) and \(\mathbf { j }\) are perpendicular unit vectors in a horizontal plane.] Three forces, \(( - 10 \mathbf { i } + a \mathbf { j } ) \mathrm { N } , ( b \mathbf { i } - 5 \mathbf { j } ) \mathrm { N }\) and \(( 2 a \mathbf { i } + 7 \mathbf { j } ) \mathrm { N }\), where \(a\) and \(b\) are constants, act on a particle \(P\) of mass 3 kg . The acceleration of \(P\) is \(( 3 \mathbf { i } + 4 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 2 }\)

1. Find the value of \(a\) and the value of \(b\). At time \(t = 0\) seconds the speed of \(P\) is \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and at time \(t = 4\) seconds the velocity of \(P\) is \(( 20 \mathbf { i } + 20 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\)
2. Find the value of \(u\).

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 \(b\).

5.
[In this question the unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are horizontal vectors due east and due north respectively and position vectors are given relative to a fixed origin.]