1.10e Position vectors: and displacement

6

1. Quadrilateral KLMN has vertices \(\mathrm { K } ( - 4,1 ) , \mathrm { L } ( 5 , - 1 ) , \mathrm { M } ( 6,2 )\) and \(\mathrm { N } ( 2,5 )\), as shown in Fig. 6.1. \begin{figure}[h]
   \captionsetup{labelformat=empty} \caption{Fig. 6.1} \includegraphics[alt={},max width=\textwidth]{20639e13-01cc-4d96-b694-fb3cf1828f4d-06_567_1004_404_319}
   \end{figure}
   1. Find the coordinates of the following midpoints.
      • P , the midpoint of KL
      • Q, the midpoint of LM
      • R, the midpoint of MN
      • S, the midpoint of NK
      • Verify that PQRS is a parallelogram.
      • TVWX is a quadrilateral as shown in Fig. 6.2.
      Points A and B divide side TV into 3 equal parts. Points C and D divide side VW into 3 equal parts. Points E and F divide side WX into 3 equal parts. Points G and H divide side TX into 3 equal parts. \(\overrightarrow { \mathrm { TA } } = \mathbf { a } , \quad \overrightarrow { \mathrm { TH } } = \mathbf { b } , \quad \overrightarrow { \mathrm { VC } } = \mathbf { c }\). \begin{figure}[h]
      \captionsetup{labelformat=empty} \caption{Fig. 6.2} \includegraphics[alt={},max width=\textwidth]{20639e13-01cc-4d96-b694-fb3cf1828f4d-06_577_671_1877_319}
      \end{figure}
      1. Show that \(\overrightarrow { \mathrm { WX } } = k ( - \mathbf { a } + \mathbf { b } - \mathbf { c } )\), where \(k\) is a constant to be determined.
      2. Verify that AH is parallel to DE .
      3. Verify that BC is parallel to GF .

1 The line \(A B\) has equation \(7 x + 3 y = 13\).

1. Find the gradient of \(A B\).
2. The point \(C\) has coordinates \(( - 1,3 )\).
   1. Find an equation of the line which passes through the point \(C\) and which is parallel to \(A B\).
   2. The point \(\left( 1 \frac { 1 } { 2 } , - 1 \right)\) is the mid-point of \(A C\). Find the coordinates of the point \(A\).
3. The line \(A B\) intersects the line with equation \(3 x + 2 y = 12\) at the point \(B\). Find the coordinates of \(B\).

1 The point \(A\) has coordinates \(( - 1,2 )\) and the point \(B\) has coordinates \(( 3 , - 5 )\).

1. 1. Find the gradient of \(A B\).
   2. Hence find an equation of the line \(A B\), giving your answer in the form \(p x + q y = r\), where \(p , q\) and \(r\) are integers.
2. The midpoint of \(A B\) is \(M\).
   1. Find the coordinates of \(M\).
   2. Find an equation of the line which passes through \(M\) and which is perpendicular to \(A B\). [3 marks]
3. The point \(C\) has coordinates \(( k , 2 k + 3 )\). Given that the distance from \(A\) to \(C\) is \(\sqrt { 13 }\), find the two possible values of the constant \(k\).
   [0pt] [4 marks]

5 A circle with centre \(C ( 5 , - 3 )\) passes through the point \(A ( - 2,1 )\).

1. Find the equation of the circle in the form $$( x - a ) ^ { 2 } + ( y - b ) ^ { 2 } = k$$
2. Given that \(A B\) is a diameter of the circle, find the coordinates of the point \(B\).
3. Find an equation of the tangent to the circle at the point \(A\), giving your answer in the form \(p x + q y + r = 0\), where \(p , q\) and \(r\) are integers.
4. The point \(T\) lies on the tangent to the circle at \(A\) such that \(A T = 4\). Find the length of \(C T\).

6

1. The points \(A , B\) and \(C\) have coordinates \(( 3,1 , - 6 ) , ( 5 , - 2,0 )\) and \(( 8 , - 4 , - 6 )\) respectively.
   1. Show that the vector \(\overrightarrow { A C }\) is given by \(\overrightarrow { A C } = n \left[ \begin{array} { r } 1 \\ - 1 \\ 0 \end{array} \right]\), where \(n\) is an integer.
   2. Show that the acute angle \(A C B\) is given by \(\cos ^ { - 1 } \left( \frac { 5 \sqrt { 2 } } { 14 } \right)\).
2. Find a vector equation of the line \(A C\).
3. The point \(D\) has coordinates \(( 6 , - 1 , p )\). It is given that the lines \(A C\) and \(B D\) intersect.
   1. Find the value of \(p\).
   2. Show that \(A B C D\) is a rhombus, and state the length of each of its sides.

5 The points \(A\) and \(B\) have coordinates \(( 5,1 , - 2 )\) and \(( 4 , - 1,3 )\) respectively.
The line \(l\) has equation \(\mathbf { r } = \left[ \begin{array} { r } - 8 \\ 5 \\ - 6 \end{array} \right] + \mu \left[ \begin{array} { r } 5 \\ 0 \\ - 2 \end{array} \right]\).

1. Find a vector equation of the line that passes through \(A\) and \(B\).
2. 1. Show that the line that passes through \(A\) and \(B\) intersects the line \(l\), and find the coordinates of the point of intersection, \(P\).
   2. The point \(C\) lies on \(l\) such that triangle \(P B C\) has a right angle at \(B\). Find the coordinates of \(C\).

\(\mathbf { 7 } \quad\) The line \(l _ { 1 }\) has equation \(\mathbf { r } = \left[ \begin{array} { r } 0 \\ - 2 \\ q \end{array} \right] + \lambda \left[ \begin{array} { r } 2 \\ 0 \\ - 1 \end{array} \right]\), where \(q\) is an integer. The line \(l _ { 2 }\) has equation \(\mathbf { r } = \left[ \begin{array} { l } 8 \\ 3 \\ 5 \end{array} \right] + \mu \left[ \begin{array} { l } 2 \\ 5 \\ 4 \end{array} \right]\). The lines \(l _ { 1 }\) and \(l _ { 2 }\) intersect at the point \(P\).

1. Show that \(q = 4\) and find the coordinates of \(P\).
2. Show that \(l _ { 1 }\) and \(l _ { 2 }\) are perpendicular.
3. The point \(A\) lies on the line \(l _ { 1 }\) where \(\lambda = 1\).
   1. Find \(A P ^ { 2 }\).
   2. The point \(B\) lies on the line \(l _ { 2 }\) so that the right-angled triangle \(A P B\) is isosceles. Find the coordinates of the two possible positions of \(B\).

6 The points \(A , B\) and \(C\) have coordinates \(( 3 , - 2,4 ) , ( 1 , - 5,6 )\) and \(( - 4,5 , - 1 )\) respectively. The line \(l\) passes through \(A\) and has equation \(\mathbf { r } = \left[ \begin{array} { r } 3 \\ - 2 \\ 4 \end{array} \right] + \lambda \left[ \begin{array} { r } 7 \\ - 7 \\ 5 \end{array} \right]\).

1. Show that the point \(C\) lies on the line \(l\).
2. Find a vector equation of the line that passes through points \(A\) and \(B\).
3. The point \(D\) lies on the line through \(A\) and \(B\) such that the angle \(C D A\) is a right angle. Find the coordinates of \(D\).
4. The point \(E\) lies on the line through \(A\) and \(B\) such that the area of triangle \(A C E\) is three times the area of triangle \(A C D\). Find the coordinates of the two possible positions of \(E\).

5. Relative to a fixed origin, the points \(A , B\) and \(C\) have position vectors ( \(2 \mathbf { i } - \mathbf { j } + 6 \mathbf { k }\) ), \(( 5 \mathbf { i } - 4 \mathbf { j } )\) and \(( 7 \mathbf { i } - 6 \mathbf { j } - 4 \mathbf { k } )\) respectively.

1. Show that \(A , B\) and \(C\) all lie on a single straight line.
2. Write down the ratio \(A B : B C\) The point \(D\) has position vector \(( 3 \mathbf { i } + \mathbf { j } + 4 \mathbf { k } )\).
3. Show that \(A D\) is perpendicular to \(B D\).
4. Find the exact area of triangle \(A B D\).
   5. continued

7 The unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are directed east and north respectively. A yacht moves with a constant acceleration. At time \(t\) seconds the position vector of the yacht is \(\mathbf { r }\) metres. When \(t = 0\) the velocity of the yacht is \(( 2 \mathbf { i } - \mathbf { j } ) \mathrm { ms } ^ { - 1 }\), and when \(t = 10\) the velocity of the yacht is \(( - \mathbf { i } + \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\).

1. Find the acceleration of the yacht.
2. When \(t = 0\) the yacht is 20 metres due east of the origin. Find an expression for \(\mathbf { r }\) in terms of \(t\).
3. 1. Show that when \(t = 20\) the yacht is due north of the origin.
   2. Find the speed of the yacht when \(t = 20\).

8 A particle is initially at the origin, where it has velocity \(( 5 \mathbf { i } - 2 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\). It moves with a constant acceleration \(\mathbf { a } \mathrm { ms } ^ { - 2 }\) for 10 seconds to the point with position vector \(75 \mathbf { i }\) metres.

1. Show that \(\mathbf { a } = 0.5 \mathbf { i } + 0.4 \mathbf { j }\).
2. Find the position vector of the particle 8 seconds after it has left the origin.
3. Find the position vector of the particle when it is travelling parallel to the unit vector \(\mathbf { i }\).

8 A Jet Ski is at the origin and is travelling due north at \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) when it begins to accelerate uniformly. After accelerating for 40 seconds, it is travelling due east at \(4 \mathrm {~ms} ^ { - 1 }\). The unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are directed east and north respectively.

1. Show that the acceleration of the Jet Ski is \(( 0.1 \mathbf { i } - 0.125 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 2 }\).
2. Find the position vector of the Jet Ski at the end of the 40 second period.
3. The Jet Ski is travelling southeast \(t\) seconds after it leaves the origin.
   1. Find \(t\).
   2. Find the velocity of the Jet Ski at this time.

7 A particle is initially at the point \(A\), which has position vector \(13.6 \mathbf { i }\) metres, with respect to an origin \(O\). At the point \(A\), the particle has velocity \(( 6 \mathbf { i } + 2.4 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\), and in its subsequent motion, it has a constant acceleration of \(( - 0.8 \mathbf { i } + 0.1 \mathbf { j } ) \mathrm { ms } ^ { - 2 }\). The unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are directed east and north respectively.

1. Find an expression for the velocity of the particle \(t\) seconds after it leaves \(A\).
2. Find an expression for the position vector of the particle, with respect to the origin \(O\), \(t\) seconds after it leaves \(A\).
3. Find the distance of the particle from the origin \(O\) when it is travelling in a north-westerly direction.
   \includegraphics[max width=\textwidth, alt={}]{ccc1db66-9700-4f22-905e-cc0bdf1fd3c1-17_2486_1709_221_153}

7 A particle moves on a smooth horizontal plane. It is initially at the point \(A\), with position vector \(( 9 \mathbf { i } + 7 \mathbf { j } ) \mathrm { m }\), and has velocity \(( - 2 \mathbf { i } + 2 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\). The particle moves with a constant acceleration of \(( 0.25 \mathbf { i } + 0.3 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 2 }\) for 20 seconds until it reaches the point \(B\). The unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are directed east and north respectively.

1. Find the velocity of the particle at the point \(B\).
2. Find the velocity of the particle when it is travelling due north.
3. Find the position vector of the point \(B\).
4. Find the average velocity of the particle as it moves from \(A\) to \(B\).
   
   \includegraphics[max width=\textwidth, alt={}]{c022c936-72bc-4cf9-8f98-285f12c1d479-15_2484_1709_223_153}

7 A particle, of mass 10 kg , moves on a smooth horizontal surface. A single horizontal force, \(( 9 \mathbf { i } + 12 \mathbf { j } )\) newtons, acts on the particle. The unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are directed east and north respectively.

1. Find the acceleration of the particle.
2. At time \(t\) seconds, the velocity of the particle is \(\mathbf { v } \mathrm { m } \mathrm { s } ^ { - 1 }\). When \(t = 0\), the velocity of the particle is \(( 2.2 \mathbf { i } + \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\) and the particle is at the origin.
   1. Find the distance between the particle and the origin when \(t = 5\).
   2. Express \(\mathbf { v }\) in terms of \(t\).
   3. Find \(t\) when the particle is travelling north-east.
      \includegraphics[max width=\textwidth, alt={}]{5d474771-fe32-47c6-8bf3-60ff7a25dd12-15_2484_1709_223_153}

5. Two dogs, Fido and Growler, are playing in a field. Fido is moving in a straight line so that at time \(t\) his position vector relative to a fixed origin, \(O\), is given by \([ ( 2 t - 3 ) \mathbf { i } + t \mathbf { j } ]\) metres. Growler is stationary at the point with position vector \(( 2 \mathbf { i } + 5 \mathbf { j } )\) metres, where \(\mathbf { i }\) and \(\mathbf { j }\) are horizontal perpendicular unit vectors.

1. Find the displacement vector of Fido from Growler in terms of \(t\).
2. Find the value of \(t\) for which the two dogs are closest.
3. Find the minimum distance between the two dogs.

4. The position of an aeroplane flying in a straight horizontal line at constant speed is plotted on a radar screen. At 2 p.m. the position vector of the aeroplane is \(( 80 \mathbf { i } + 5 \mathbf { j } )\), where \(\mathbf { i }\) and \(\mathbf { j }\) are unit vectors directed east and north respectively relative to a fixed origin, \(O\), on the screen. Ten minutes later the position of the aeroplane on the screen is \(( 32 \mathbf { i } + 19 \mathbf { j } )\). Each unit on the screen represents 1 km .

1. Find the position vector of the aeroplane at 2:30 p.m.
2. Find the speed of the aeroplane in \(\mathrm { km } \mathrm { h } ^ { - 1 }\).
3. Find, correct to the nearest degree, the bearing on which the aeroplane is flying.

7. At 6 a.m. a cargo ship has position vector \(( 7 \mathbf { i } + 56 \mathbf { j } ) \mathrm { km }\) relative to a fixed origin \(O\) on the coast and moves with constant velocity \(( 9 \mathbf { i } - 6 \mathbf { j } ) \mathrm { kmh } ^ { - 1 }\). A ferry sails from \(O\) at 6 a.m. and moves with constant velocity \(( 12 \mathbf { i } + 18 \mathbf { j } ) \mathrm { km } \mathrm { h } ^ { - 1 }\). The unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are directed due east and due north respectively.

1. Show that the position vector of the cargo ship \(t\) hours after 6 a.m. is given by $$[ ( 7 + 9 t ) \mathbf { i } + ( 56 - 6 t ) \mathbf { j } ] \mathrm { km }$$ and find the position vector of the ferry in terms of \(t\).
2. Show that if both vessels maintain their course and speed, they will collide and find the time and position vector at which this occurs.
   (6 marks)
   At 8 a.m. the captain of the ferry realises that a collision is imminent and changes course so that the ferry now has velocity \(( 21 \mathbf { i } + 6 \mathbf { j } ) \mathrm { kmh } ^ { - 1 }\).
3. Find the distance between the two ships at the time when they would have collided.

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.

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, 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.

1 The vectors \(\mathbf { P } , \mathbf { Q }\) and \(\mathbf { R }\) are given by $$\mathbf { P } = 5 \mathbf { i } + 4 \mathbf { j } , \quad \mathbf { Q } = 3 \mathbf { i } - 5 \mathbf { j } , \quad \mathbf { R } = - 8 \mathbf { i } + \mathbf { j }$$

1. Find the vector \(\mathbf { P } + \mathbf { Q } + \mathbf { R }\).
2. Interpret your answer to part (i) in the cases
   (A) \(\mathbf { P } , \mathbf { Q }\) and \(\mathbf { R }\) represent three forces acting on a particle,
   (B) \(\mathbf { P } , \mathbf { Q }\) and \(\mathbf { R }\) represent three stages of a hiker's walk.

2 The vectors \(\mathbf { P } , \mathbf { Q }\) and \(\mathbf { R }\) are given by $$\mathbf { P } = 5 \mathbf { i } + 4 \mathbf { j } , \quad \mathbf { Q } = 3 \mathbf { i } - 5 \mathbf { j } , \quad \mathbf { R } = - 8 \mathbf { i } + \mathbf { j }$$

1. Find the vector \(\mathbf { P } + \mathbf { Q } + \mathbf { R }\).
2. Interpret your answer to part (i) in the cases
   (A) \(\mathbf { P } , \mathbf { Q }\) and \(\mathbf { R }\) represent three forces acting on a particle,
   (B) \(\mathbf { P } , \mathbf { Q }\) and \(\mathbf { R }\) represent three stages of a hiker's walk.

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.

2 The position vector of a particle at time \(t\) is given by $$\mathbf { r } = \frac { 1 } { 2 } t \mathbf { i } + \left( t ^ { 2 } - 1 \right) \mathbf { j } .$$ referred to an origin \(O\) where \(\mathbf { i }\) and \(\mathbf { j }\) are the standard unit vectors in the directions of the cartesian axes Ox and Oy respectively.

1. Write down the value of \(t\) for which the \(x\)-coordinate of the position of the particle is 2 . Find the \(y\)-coordinate at this time.
2. Show that the cartesian equation of the path of the particle is \(y = 4 x ^ { 2 } - 1\).
3. Find the coordinates of the point where the particle is moving at \(45 ^ { \circ }\) to both Ox and Oy .