1.10c Magnitude and direction: of vectors

1. At time \(t = 0\), a particle of mass 2 kg has velocity \(( 8 \mathbf { i } + \lambda \mathbf { j } ) \mathrm { ms } ^ { - 1 }\) where \(\mathbf { i }\) and \(\mathbf { j }\) are horizontal perpendicular unit vectors and \(\lambda > 0\).

Given that the speed of the particle at time \(t = 0\) is \(17 \mathrm {~m} \mathrm {~s} ^ { - 1 }\),

1. find the value of \(\lambda\). The particle experiences a constant retarding force \(\mathbf { F }\) so that when \(t = 5\), it has velocity \(( 3 \mathbf { i } + 5 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\).
2. Show that \(\mathbf { F }\) can be written in the form \(\mu ( \mathbf { i } + 2 \mathbf { j } ) \mathrm { N }\) where \(\mu\) is a constant which you should find.
   (5 marks)

1 The map of a large area of open land is marked in 1 km squares and a point near the middle of the area is defined to be the origin. The vectors \(\binom { 1 } { 0 }\) and \(\binom { 0 } { 1 }\) are in the directions east and north. At time \(t\) hours the position vectors of two hikers, Ashok and Kumar, are given by: $$\begin{array} { l l } \text { Ashok } & \mathbf { r } _ { \mathrm { A } } = \binom { - 2 } { 0 } + \binom { 8 } { 1 } t , \\ \text { Kumar } & \mathbf { r } _ { \mathrm { K } } = \binom { 7 t } { 10 - 4 t } . \end{array}$$

1. Prove that the two hikers meet and give the coordinates of the point where this happens.
2. Compare the speeds of the two hikers.

3 In this question the origin is a point on the ground. The directions of the unit vectors \(\left( \begin{array} { l } 1 \\ 0 \\ 0 \end{array} \right) , \left( \begin{array} { l } 0 \\ 1 \\ 0 \end{array} \right)\) and \(\left( \begin{array} { l } 0 \\ 0 \\ 1 \end{array} \right)\) are
east, north and vertically upwards. \includegraphics[max width=\textwidth, alt={}, center]{cb72a1c4-f769-4348-ad7f-66c3c96e1732-3_401_686_368_721} Alesha does a sky-dive on a day when there is no wind. The dive starts when she steps out of a moving helicopter. The dive ends when she lands gently on the ground.

• During the dive Alesha can reduce the magnitude of her acceleration in the vertical direction by spreading her arms and increasing air resistance.
• During the dive she can use a power unit strapped to her back to give herself an acceleration in a horizontal direction.
• Alesha's mass, including her equipment, is 100 kg .
• Initially, her position vector is \(\left( \begin{array} { r } - 75 \\ 90 \\ 750 \end{array} \right) \mathrm { m }\) and her velocity is \(\left( \begin{array} { r } - 5 \\ 0 \\ - 10 \end{array} \right) \mathrm { ms } ^ { - 1 }\).
  1. Calculate Alesha's initial speed, and the initial angle between her motion and the downward vertical.

At a certain time during the dive, forces of \(\left( \begin{array} { r } 0 \\ 0 \\ - 980 \end{array} \right) \mathrm { N } , \left( \begin{array} { r } 0 \\ 0 \\ 880 \end{array} \right) \mathrm { N }\) and \(\left( \begin{array} { r } 50 \\ - 20 \\ 0 \end{array} \right) \mathrm { N }\) are acting on Alesha.

Suggest how these forces could arise.

Find Alesha's acceleration at this time, giving your answer in vector form, and show that, correct to 3 significant figures, its magnitude is \(1.14 \mathrm {~ms} ^ { - 2 }\). One suggested model for Alesha's motion is that the forces on her are constant throughout the dive from when she leaves the helicopter until she reaches the ground.

Find expressions for her velocity and position vector at time \(t\) seconds after the start of the dive according to this model. Verify that when \(t = 30\) she is at the origin.

Explain why consideration of Alesha's landing velocity shows this model to be unrealistic.

5 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.

2 In this question, air resistance should be neglected.
Fig. 2 illustrates the flight of a golf ball. The golf ball is initially on the ground, which is horizontal. \begin{figure}[h] \end{figure} It is hit and given an initial velocity with components of \(15 \mathrm {~ms} ^ { - 1 }\) in the horizontal direction and \(20 \mathrm {~ms} ^ { - 1 }\) in the vertical direction.

1. Find its initial speed.
2. Find the ball's flight time and range, \(R \mathrm {~m}\).
3. (A) Show that the range is the same if the components of the initial velocity of the ball are \(20 \mathrm {~ms} ^ { - 1 }\) in the horizontal direction and \(15 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in the vertical direction.
   (B) State, justifying your answer, whether the range is the same whenever the ball is hit with the same initial speed.

1 In this question take \(\boldsymbol { g } = \mathbf { 1 0 }\).
The directions of the unit vectors \(\left( \begin{array} { l } 1 \\ 0 \\ 0 \end{array} \right) , \left( \begin{array} { l } 0 \\ 1 \\ 0 \end{array} \right)\) and \(\left( \begin{array} { l } 0 \\ 0 \\ 1 \end{array} \right)\) are east, north and vertically upwards.
Forces \(\mathbf { p } , \mathbf { q }\) and \(\mathbf { r }\) are given by \(\mathbf { p } = \left( \begin{array} { r } - 1 \\ - 1 \\ 5 \end{array} \right) \mathrm { N } , \mathbf { q } = \left( \begin{array} { r } - 1 \\ - 4 \\ 2 \end{array} \right) \mathrm { N }\) and \(\mathbf { r } = \left( \begin{array} { l } 2 \\ 5 \\ 0 \end{array} \right) \mathrm { N }\).

1. Find which of \(\mathbf { p } , \mathbf { q }\) and \(\mathbf { r }\) has the greatest magnitude.
2. A particle has mass 0.4 kg . The forces acting on it are \(\mathbf { p } , \mathbf { q } , \mathbf { r }\) and its weight. Find the magnitude of the particle's acceleration and describe the direction of this acceleration.

1 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 Force \(\mathbf { F } _ { 1 }\) is \(\binom { 6 } { 13 } \mathrm {~N}\) and force \(\mathbf { F } _ { 2 }\) is \(\binom { 3 } { 5 }\), where \({ } _ { 0 }\) and \(\binom { 0 } { 1 }\) are vectors east and north respectively.

1. Calculate the magnitude of \(\mathbf { F } _ { 1 }\), correct to three significant figures.
2. Calculate the direction of the force \(\mathbf { F } _ { 1 } - \mathbf { F } _ { 2 }\) as a bearing. Force \(\mathbf { F } _ { 2 }\) is the resultant of all the forces acting on an object of mass 5 kg .
3. Calculate the acceleration of the object and the change in its velocity after 10 seconds.

2 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.

3 The trajectory ABCD of a small stone moving with negligible air resistance is shown in Fig. 7. AD is horizontal and BC is parallel to AD . The stone is projected from A with speed \(40 \mathrm {~ms} ^ { - 1 }\) at \(50 ^ { \circ }\) to the horizontal. \begin{figure}[h] \end{figure}

1. Write down an expression for the horizontal displacement from A of the stone \(t\) seconds after projection. Write down also an expression for the vertical displacement at time \(t\).
2. Show that the stone takes 6.253 seconds (to three decimal places) to travel from A to D . Calculate the range of the stone. You are given that \(X = 30\).
3. Calculate the time it takes the stone to reach B . Hence determine the time for it to travel from A to C.
4. Calculate the direction of the motion of the stone at C .

3 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 \(k\).

4 A small box has weight \(\mathbf { W } \mathrm { N }\) and is held in equilibrium by two strings with tensions \(\mathbf { T } _ { 1 } \mathrm {~N}\) and \(\mathbf { T } _ { 2 } \mathrm {~N}\). This situation is shown in Fig. 2 which also shows the standard unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) that are horizontal and vertically upwards, respectively. \begin{figure}[h] \end{figure} The tension \(\mathbf { T } _ { 1 }\) is \(10 \mathbf { i } + 24 \mathbf { j }\).

1. Calculate the magnitude of \(\mathbf { T } _ { 1 }\) and the angle between \(\mathbf { T } _ { 1 }\) and the vertical. The magnitude of the weight is \(w \mathrm {~N}\).
2. Write down the vector \(\mathbf { W }\) in terms of \(w\) and \(\mathbf { j }\). The tension \(\mathbf { T } _ { 2 }\) is \(k \mathbf { i } + 10 \mathbf { j }\), where \(k\) is a scalar.
3. Find the values of \(k\) and of \(w\).

5 A particle has a position vector \(\mathbf { r }\), where \(\mathbf { r } = 4 \mathbf { i } - 5 \mathbf { j }\) and \(\mathbf { i }\) and \(\mathbf { j }\) are unit vectors in the directions east and north respectively.

1. Sketch \(\mathbf { r }\) on a diagram showing \(\mathbf { i }\) and \(\mathbf { j }\) and the origin O .
2. Calculate the magnitude of \(\mathbf { r }\) and its direction as a bearing.
3. Write down the vector that has the same direction as \(\mathbf { r }\) and three times its magnitude.

6 Force \(\mathbf { F } _ { 1 }\) is \(\binom { 6 } { 13 } \mathrm {~N}\) and force \(\mathbf { F } _ { 2 }\) is \(\binom { 3 } { 5 }\), where \(\left. \int _ { 0 } \right] _ { \text {and } } \binom { 0 } { 1 }\) are vectors east and north respectively.

1. Calculate the magnitude of \(\mathbf { F } _ { 1 }\), correct to three significant figures.
2. Calculate the direction of the force \(\mathbf { F } _ { 1 } - \mathbf { F } _ { 2 }\) as a bearing. Force \(\mathbf { F } _ { 2 }\) is the resultant of all the forces acting on an object of mass 5 kg .
3. Calculate the acceleration of the object and the change in its velocity after 10 seconds.

1 A particle rests on a smooth, horizontal plane. Horizontal unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) lie in this plane. The particle is in equilibrium under the action of the three forces \(( - 3 \mathbf { i } + 4 \mathbf { j } ) \mathrm { N }\) and \(( 21 \mathbf { i } - 7 \mathbf { j } ) \mathrm { N }\) and \(\mathbf { R N }\).

1. Write down an expression for \(\mathbf { R }\) in terms of \(\mathbf { i }\) and \(\mathbf { j }\).
2. Find the magnitude of \(\mathbf { R }\) and the angle between \(\mathbf { R }\) and the \(\mathbf { i }\) direction.

3 The vectors \(\mathbf { p }\) and \(\mathbf { q }\) are given by $$\mathbf { p } = 8 \mathbf { i } + \mathbf { j } \text { and } \mathbf { q } = 4 \mathbf { i } - 7 \mathbf { j } .$$

1. Show that \(\mathbf { p }\) and \(\mathbf { q }\) are equal in magnitude.
2. Show that \(\mathbf { p } + \mathbf { q }\) is parallel to \(2 \mathbf { i } - \mathbf { j }\).
3. Draw \(\mathbf { p } + \mathbf { q }\) and \(\mathbf { p } - \mathbf { q }\) on the grid. Write down the angle between these two vectors.

4 In this question, \(\mathbf { i }\) is a horizontal unit vector and \(\mathbf { j }\) is a unit vector pointing vertically upwards.
A force \(\mathbf { F }\) is \(- \mathbf { i } + 5 \mathbf { j }\).

1. Calculate the magnitude of \(\mathbf { F }\). Calculate also the angle between \(\mathbf { F }\) and the upward vertical. Force \(\mathbf { G }\) is \(2 a \mathbf { i } + a \mathbf { j }\) and force \(\mathbf { H }\) is \(- 2 \mathbf { i } + 3 b \mathbf { j }\), where \(a\) and \(b\) are constants. The force \(\mathbf { H }\) is the resultant of forces \(4 \mathbf { F }\) and \(\mathbf { G }\).
2. Find \(\mathbf { G }\) and \(\mathbf { H }\).

5 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 \(\binom { 2 } { 3 } \mathrm {~m} \mathrm {~s} ^ { - 2 }\).

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

6 The force acting on a particle of mass 1.5 kg is given by the vector \(\binom { 6 } { 9 } \mathrm {~N}\).

1. Give the acceleration of the particle as a vector.
2. Calculate the angle that the acceleration makes with the direction \(\binom { 1 } { 0 }\).
3. At a certain point of its motion, the particle has a velocity of \(\binom { - 2 } { 3 } \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Calculate the displacement of the particle over the next two seconds.

7 A force \(\mathbf { F }\) is given by \(\mathbf { F } = ( 3.5 \mathbf { i } + 12 \mathbf { j } ) \mathrm { N }\), where \(\mathbf { i }\) and \(\mathbf { j }\) are horizontal unit vectors east and north respectively.

1. Calculate the magnitude of \(\mathbf { F }\) and also its direction as a bearing.
2. \(\mathbf { G }\) is the force \(( 7 \mathbf { i } + 24 \mathbf { j } ) \mathrm { N }\). Show that \(\mathbf { G }\) and \(\mathbf { F }\) are in the same direction and compare their magnitudes.
3. Force \(\mathbf { F } _ { 1 }\) is \(( 9 \mathbf { i } - 18 \mathbf { j } ) \mathrm { N }\) and force \(\mathbf { F } _ { 2 }\) is \(( 12 \mathbf { i } + q \mathbf { j } ) \mathrm { N }\). Find \(q\) so that the sum \(\mathbf { F } _ { 1 } + \mathbf { F } _ { 2 }\) is in the direction of \(\mathbf { F }\).

1 A particle moves in a horizontal plane, in which the unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are directed east and north respectively. At time \(t\) seconds, its position vector, \(\mathbf { r }\) metres, is given by $$\mathbf { r } = \left( 2 t ^ { 3 } - t ^ { 2 } + 6 \right) \mathbf { i } + \left( 8 - 4 t ^ { 3 } + t \right) \mathbf { j }$$

1. Find an expression for the velocity of the particle at time \(t\).
2. 1. Find the velocity of the particle when \(t = \frac { 1 } { 3 }\).
   2. State the direction in which the particle is travelling at this time.
3. Find the acceleration of the particle when \(t = 4\).
4. The mass of the particle is 6 kg . Find the magnitude of the resultant force on the particle when \(t = 4\).

2 The unit vectors \(\mathbf { i } , \mathbf { j }\) and \(\mathbf { k }\) are directed due east, due north and vertically upwards respectively. Two helicopters, \(A\) and \(B\), are flying with constant velocities of \(( 20 \mathbf { i } - 10 \mathbf { j } + 20 \mathbf { k } ) \mathrm { ms } ^ { - 1 }\) and \(( 30 \mathbf { i } + 10 \mathbf { j } + 10 \mathbf { k } ) \mathrm { m } \mathrm { s } ^ { - 1 }\) respectively. At noon, the position vectors of \(A\) and \(B\) relative to a fixed origin, \(O\), are \(( 8000 \mathbf { i } + 1500 \mathbf { j } + 3000 \mathbf { k } ) \mathrm { m }\) and \(( 2000 \mathbf { i } + 500 \mathbf { j } + 1000 \mathbf { k } ) \mathrm { m }\) respectively.

1. Write down the velocity of \(A\) relative to \(B\).
2. Find the position vector of \(A\) relative to \(B\) at time \(t\) seconds after noon.
3. Find the value of \(t\) when \(A\) and \(B\) are closest together.

4 Two smooth spheres, \(A\) and \(B\), have equal radii and masses \(m\) and \(2 m\) respectively. The spheres are moving on a smooth horizontal plane. The sphere \(A\) has velocity ( \(4 \mathbf { i } + 3 \mathbf { j }\) ) when it collides with the sphere \(B\) which has velocity \(( - 2 \mathbf { i } + 2 \mathbf { j } )\). After the collision, the velocity of \(B\) is \(( \mathbf { i } + \mathbf { j } )\).

1. Find the velocity of \(A\) immediately after the collision.
2. Find the angle between the velocities of \(A\) and \(B\) immediately after the collision.
3. Find the impulse exerted by \(B\) on \(A\).
4. State, as a vector, the direction of the line of centres of \(A\) and \(B\) when they collide.
   (1 mark)

4 The unit vectors \(\mathbf { i } , \mathbf { j }\) and \(\mathbf { k }\) are directed due east, due north and vertically upwards respectively. A helicopter, \(A\), is travelling in the direction of the vector \(- 2 \mathbf { i } + 3 \mathbf { j } + 6 \mathbf { k }\) with constant speed \(140 \mathrm {~km} \mathrm {~h} ^ { - 1 }\). Another helicopter, \(B\), is travelling in the direction of the vector \(2 \mathbf { i } - \mathbf { j } + 2 \mathbf { k }\) with constant speed \(60 \mathrm {~km} \mathrm {~h} ^ { - 1 }\).

1. Find the velocity of \(A\) relative to \(B\).
2. Initially, the position vectors of \(A\) and \(B\) are \(( 4 \mathbf { i } - 2 \mathbf { j } + 3 \mathbf { k } ) \mathrm { km }\) and \(( - 3 \mathbf { i } + 6 \mathbf { j } + 3 \mathbf { k } ) \mathrm { km }\) respectively, relative to a fixed origin. Write down the position vector of \(A\) relative to \(B , t\) hours after they leave their initial positions.
3. Find the distance between \(A\) and \(B\) when they are closest together.
   
   \includegraphics[max width=\textwidth, alt={}]{0590950d-145c-4ae2-bc3c-f61a9433d158-10_2486_1714_221_153}
   \includegraphics[max width=\textwidth, alt={}]{0590950d-145c-4ae2-bc3c-f61a9433d158-11_2486_1714_221_153}

6 At noon, two ships, \(A\) and \(B\), are a distance of 12 km apart, with \(B\) on a bearing of \(065 ^ { \circ }\) from \(A\). The ship \(B\) travels due north at a constant speed of \(10 \mathrm {~km} \mathrm {~h} ^ { - 1 }\). The ship \(A\) travels at a constant speed of \(18 \mathrm {~km} \mathrm {~h} ^ { - 1 }\). \includegraphics[max width=\textwidth, alt={}, center]{a90a2de3-5cc0-4e87-b29a-2562f86eee17-16_492_585_445_738}

1. Find the direction in which \(A\) should travel in order to intercept \(B\). Give your answer as a bearing.
2. In fact, the ship \(A\) actually travels on a bearing of \(065 ^ { \circ }\).
   1. Find the distance between the ships when they are closest together.
   2. Find the time when the ships are closest together.