1.10e Position vectors: and displacement

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1. (ii) Verify that \(k = 0.8\) [0pt] [1 mark] \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) 19
2. Find the position vector of Amba when \(t = 4\) [0pt] [3 marks] \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) 19
3. At both \(t = 0\) and \(t = 4\) there is a distance of 5 metres between Jo and Amba's positions. Determine the shortest distance between their two parallel lines of motion.
   Fully justify your answer. \includegraphics[max width=\textwidth, alt={}, center]{b7df05bf-f3fc-4705-a13c-6b562896fa9f-32_2492_1721_217_150}

6 A circle has centre \(C ( 10,4 )\). The \(x\)-axis is a tangent to the circle, as shown in Fig. 6. \begin{figure}[h] \end{figure}

1. Find the equation of the circle.
2. Show that the line \(y = x\) is not a tangent to the circle.
3. Write down the position vector of the midpoint of OC.

1. \hspace{0pt} [In this question \(\mathbf { i }\) and \(\mathbf { j }\) are horizontal unit vectors and position vectors are given relative to a fixed origin.]

A ship \(A\) is moving with constant velocity.
At 1 pm , the position vector of \(A\) is \(( 25 \mathbf { i } + 10 \mathbf { j } ) \mathrm { km }\).
At 3 pm , the position vector of \(A\) is \(( 55 \mathbf { i } + 34 \mathbf { j } ) \mathrm { km }\).
At time \(t\) hours after 1 pm , the position vector of \(A\) is \(\mathbf { r } _ { A } \mathrm {~km}\).

1. Show that \(\mathbf { r } _ { A } = ( 25 + 15 t ) \mathbf { i } + ( 10 + 12 t ) \mathbf { j }\) The speed of \(A\) is \(V \mathrm {~ms} ^ { - 1 }\)
2. Find the value of \(V\). A ship \(B\) is moving with constant velocity \(( 20 \mathbf { i } - 6 \mathbf { j } ) \mathrm { km } \mathrm { h } ^ { - 1 }\) At 1 pm , the position vector of \(B\) is \(( 35 \mathbf { i } + 51 \mathbf { j } ) \mathrm { km }\).
   At 2:30 pm, \(B\) passes through the point \(P\).
3. Show that \(A\) also passes through \(P\).

1. Relative to a fixed origin \(O\)

• the point \(A\) has coordinates \(( - 10,5 , - 4 )\)
• the point \(B\) has coordinates \(( - 6,4 , - 1 )\)

The straight line \(l _ { 1 }\) passes through \(A\) and \(B\).

1. Find a vector equation for \(l _ { 1 }\) The line \(l _ { 2 }\) has equation $$\mathbf { r } = \left( \begin{array} { l } 3 \\ p \\ q \end{array} \right) + \mu \left( \begin{array} { r } 3 \\ - 4 \\ 1 \end{array} \right)$$ where \(p\) and \(q\) are constants and \(\mu\) is a scalar parameter.
   Given that \(l _ { 1 }\) and \(l _ { 2 }\) intersect at \(B\),
2. find the value of \(p\) and the value of \(q\). The acute angle between \(l _ { 1 }\) and \(l _ { 2 }\) is \(\theta\)
3. Find the exact value of \(\cos \theta\) Given that the point \(C\) lies on \(l _ { 2 }\) such that \(A C\) is perpendicular to \(l _ { 2 }\)
4. find the exact length of \(A C\), giving your answer as a surd.

Relative to an origin \(O\), the position vectors of points \(A\) and \(B\) are given by $$\overrightarrow{OA} = 2\mathbf{i} + 4\mathbf{j} + 4\mathbf{k} \quad \text{and} \quad \overrightarrow{OB} = 3\mathbf{i} + \mathbf{j} + 4\mathbf{k}.$$

1. Use a vector method to find angle \(AOB\). [4]

The point \(C\) is such that \(\overrightarrow{AB} = \overrightarrow{BC}\).

1. Find the unit vector in the direction of \(\overrightarrow{OC}\). [4]
2. Show that triangle \(OAC\) is isosceles. [1]

Relative to an origin \(O\), the position vectors of points \(A\) and \(B\) are given by $$\overrightarrow{OA} = \begin{pmatrix} 5 \\ 1 \\ 3 \end{pmatrix} \text{ and } \overrightarrow{OB} = \begin{pmatrix} 5 \\ 4 \\ -3 \end{pmatrix}$$ The point \(P\) lies on \(AB\) and is such that \(\overrightarrow{AP} = \frac{3}{4}\overrightarrow{AB}\).

1. Find the position vector of \(P\). [3]
2. Find the distance \(OP\). [1]
3. Determine whether \(OP\) is perpendicular to \(AB\). Justify your answer. [2]

Relative to an origin \(O\), the position vectors of points \(A\) and \(B\) are \(\mathbf{3i} + 4\mathbf{j} - \mathbf{k}\) and \(5\mathbf{i} - 2\mathbf{j} - 3\mathbf{k}\) respectively.

1. Use a scalar product to find angle \(BOA\). [4]

The point \(C\) is the mid-point of \(AB\). The point \(D\) is such that \(\overrightarrow{OD} = 2\overrightarrow{OB}\).

1. Find \(\overrightarrow{DC}\). [4]

\includegraphics{figure_7} The diagram shows a solid cylinder standing on a horizontal circular base with centre \(O\) and radius \(4\) units. Points \(A\), \(B\) and \(C\) lie on the circumference of the base such that \(AB\) is a diameter and angle \(BOC = 90°\). Points \(P\), \(Q\) and \(R\) lie on the upper surface of the cylinder vertically above \(A\), \(B\) and \(C\) respectively. The height of the cylinder is \(12\) units. The mid-point of \(CR\) is \(M\) and \(N\) lies on \(BQ\) with \(BN = 4\) units. Unit vectors \(\mathbf{i}\) and \(\mathbf{j}\) are parallel to \(OB\) and \(OC\) respectively and the unit vector \(\mathbf{k}\) is vertically upwards. Evaluate \(\overrightarrow{PN} \cdot \overrightarrow{PM}\) and hence find angle \(MPN\). [7]

With respect to the origin \(O\), the points \(A\), \(B\) and \(C\) have position vectors given by $$\overrightarrow{OA} = \begin{pmatrix} 2 \\ 1 \\ -3 \end{pmatrix}, \quad \overrightarrow{OB} = \begin{pmatrix} 0 \\ 4 \\ 1 \end{pmatrix} \quad \text{and} \quad \overrightarrow{OC} = \begin{pmatrix} -3 \\ -2 \\ 2 \end{pmatrix}.$$

1. The point \(D\) is such that \(ABCD\) is a trapezium with \(\overrightarrow{DC} = 3\overrightarrow{AB}\). Find the position vector of \(D\). [2]
2. The diagonals of the trapezium intersect at the point \(P\). Find the position vector of \(P\). [5]
3. Using a scalar product, calculate angle \(ABC\). [4]

With respect to a fixed origin, \(O\), the point \(A\) has position vector $$\overrightarrow{OA} = \begin{pmatrix} 7 \\ 2 \\ -5 \end{pmatrix}$$ Given that $$\overrightarrow{AB} = \begin{pmatrix} -2 \\ 4 \\ 3 \end{pmatrix}$$

1. find the coordinates of the point \(B\). [2]

The point \(C\) has position vector $$\overrightarrow{OC} = \begin{pmatrix} a \\ 5 \\ -1 \end{pmatrix}$$ where \(a\) is a constant. Given that \(\overrightarrow{OC}\) is perpendicular to \(\overrightarrow{BC}\)

1. find the possible values of \(a\). [4]

With respect to a fixed origin \(O\), the line \(l_1\) is given by the equation $$\mathbf{r} = \mathbf{i} + 2\mathbf{j} + 5\mathbf{k} + \lambda(8\mathbf{i} - \mathbf{j} + 4\mathbf{k})$$ where \(\lambda\) is a scalar parameter. The point \(A\) lies on \(l_1\) Given that \(|\overrightarrow{OA}| = 5\sqrt{10}\)

1. show that at \(A\) the parameter \(\lambda\) satisfies $$81\lambda^2 + 52\lambda - 220 = 0$$ [3]

Hence

1. 1. show that one possible position vector for \(A\) is \(-15\mathbf{i} + 4\mathbf{j} - 3\mathbf{k}\)
   2. find the other possible position vector for \(A\). [3]

The line \(l_2\) is parallel to \(l_1\) and passes through \(O\). Given that • \(\overrightarrow{OA} = -15\mathbf{i} + 4\mathbf{j} - 3\mathbf{k}\) • point \(B\) lies on \(l_2\) where \(|\overrightarrow{OB}| = 4\sqrt{10}\)

1. find the area of triangle \(OAB\), giving your answer to one decimal place. [4]

[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 \(\mathbf{i}\) and \(\mathbf{j}\) are horizontal unit vectors due east and due north respectively and position vectors are given relative to a fixed origin.] At 2 pm, the position vector of ship \(P\) is \((5\mathbf{i} - 3\mathbf{j})\) km and the position vector of ship \(Q\) is \((7\mathbf{i} + 5\mathbf{j})\) km.

1. Find the distance between \(P\) and \(Q\) at 2 pm. [3]

Ship \(P\) is moving with constant velocity \((2\mathbf{i} + 5\mathbf{j})\) km h\(^{-1}\) and ship \(Q\) is moving with constant velocity \((-3\mathbf{i} - 15\mathbf{j})\) km h\(^{-1}\).

1. Find the position vector of \(P\) at time \(t\) hours after 2 pm. [2]
2. Find the position vector of \(Q\) at time \(t\) hours after 2 pm. [1]
3. Show that \(Q\) will meet \(P\) and find the time at which they meet. [5]
4. Find the position vector of the point at which they meet. [2]

[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 ships, \(A\) and \(B\), are moving with constant velocities. The velocity of \(A\) is \((3\mathbf{i} + 12\mathbf{j})\text{ kmh}^{-1}\) and the velocity of \(B\) is \((p\mathbf{i} + q\mathbf{j})\text{ kmh}^{-1}\)

1. Find the speed of \(A\). [2] The ships are modelled as particles. At 12 noon, \(A\) is at the point with position vector \((-9\mathbf{i} + 6\mathbf{j})\) km and \(B\) is at the point with position vector \((16\mathbf{i} + 6\mathbf{j})\) km. At time \(t\) hours after 12 noon, $$\overrightarrow{AB} = [(25 - 12t)\mathbf{i} - 9t\mathbf{j}] \text{ km}$$
2. Find the value of \(p\) and the value of \(q\). [7]
3. Find the bearing of \(A\) from \(B\) when the ships are 15 km apart, giving your answer to the nearest degree. [7]

A particle \(P\) is moving with constant velocity \((-3\mathbf{i} + 2\mathbf{j})\) m s\(^{-1}\). At time \(t = 6\) s \(P\) is at the point with position vector \((-4\mathbf{i} - 7\mathbf{j})\) m. Find the distance of \(P\) from the origin at time \(t = 2\) s. [5]

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]

Two ships \(P\) and \(Q\) are travelling at night with constant velocities. At midnight, \(P\) is at the point with position vector \((20\mathbf{i} + 10\mathbf{j})\) km relative to a fixed origin \(O\). At the same time, \(Q\) is at the point with position vector \((14\mathbf{i} - 6\mathbf{j})\) km. Three hours later, \(P\) is at the point with position vector \((29\mathbf{i} + 34\mathbf{j})\) km. The ship \(Q\) travels with velocity \(12\mathbf{j}\) km h\(^{-1}\). At time \(t\) hours after midnight, the position vectors of \(P\) and \(Q\) are \(\mathbf{p}\) km and \(\mathbf{q}\) km respectively. Find

1. the velocity of \(P\), in terms of \(\mathbf{i}\) and \(\mathbf{j}\), [2]
2. expressions for \(\mathbf{p}\) and \(\mathbf{q}\), in terms of \(t\), \(\mathbf{i}\) and \(\mathbf{j}\). [4]

At time \(t\) hours after midnight, the distance between \(P\) and \(Q\) is \(d\) km.

1. By finding an expression for \(\overrightarrow{PQ}\), show that $$d^2 = 25t^2 - 92t + 292.$$ [5]

Weather conditions are such that an observer on \(P\) can only see the lights on \(Q\) when the distance between \(P\) and \(Q\) is 15 km or less. Given that when \(t = 1\), the lights on \(Q\) move into sight of the observer,

1. find the time, to the nearest minute, at which the lights on \(Q\) move out of sight of the observer. [5]

[In this question the horizontal unit vectors \(\mathbf{i}\) and \(\mathbf{j}\) are due east and due north respectively.] A model boat \(A\) moves on a lake with constant velocity \((-\mathbf{i} + 6\mathbf{j}) \text{ m s}^{-1}\). At time \(t = 0\), \(A\) is at the point with position vector \((2\mathbf{i} - 10\mathbf{j})\) m. Find

1. the speed of \(A\), [2]
2. the direction in which \(A\) is moving, giving your answer as a bearing. [3]

At time \(t = 0\), a second boat \(B\) is at the point with position vector \((-26\mathbf{i} + 4\mathbf{j})\) m. Given that the velocity of \(B\) is \((3\mathbf{i} + 4\mathbf{j}) \text{ m s}^{-1}\),

1. show that \(A\) and \(B\) will collide at a point \(P\) and find the position vector of \(P\). [5]

Given instead that \(B\) has speed \(8 \text{ m s}^{-1}\) and moves in the direction of the vector \((3\mathbf{i} + 4\mathbf{j})\),

1. find the distance of \(B\) from \(P\) when \(t = 7\) s. [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 along a straight line with constant velocity. At time \(t\) hours the position vector of \(S\) is \(\mathbf{s}\) km. When \(t = 0\), \(\mathbf{s} = 9\mathbf{i} - 6\mathbf{j}\). When \(t = 4\), \(\mathbf{s} = 21\mathbf{i} + 10\mathbf{j}\). Find

1. the speed of \(S\), [4]
2. the direction in which \(S\) is moving, giving your answer as a bearing. [2]
3. Show that \(\mathbf{s} = (3t + 9)\mathbf{i} + (4t - 6)\mathbf{j}\). [2]

A lighthouse \(L\) is located at the point with position vector \((18\mathbf{i} + 6\mathbf{j})\) km. When \(t = T\), the ship \(S\) is 10 km from \(L\).

1. Find the possible values of \(T\). [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 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]

A small boat \(S\), drifting in the sea, is modelled as a particle moving in a straight line at constant speed. When first sighted at 0900, \(S\) is at a point with position vector \((4\mathbf{i} - 6\mathbf{j})\) km relative to a fixed origin \(O\), where \(\mathbf{i}\) and \(\mathbf{j}\) are unit vectors due east and due north respectively. At 0945, \(S\) is at the point with position vector \((7\mathbf{i} - 7.5\mathbf{j})\) km. At time \(t\) hours after 0900, \(S\) is at the point with position vector \(\mathbf{s}\) km.

1. Calculate the bearing on which \(S\) is drifting. [4]
2. Find an expression for \(\mathbf{s}\) in terms of \(t\). [3]

At 1000 a motor boat \(M\) leaves \(O\) and travels with constant velocity \((p\mathbf{i} + q\mathbf{j})\) km h\(^{-1}\). Given that \(M\) intercepts \(S\) at 1015,

1. calculate the value of \(p\) and the value of \(q\). [6]

[In this question, the unit vectors \(\mathbf{i}\) and \(\mathbf{j}\) are horizontal vectors due east and north respectively.] At time \(t = 0\), a football player kicks a ball from the point \(A\) with position vector \((2\mathbf{i} + \mathbf{j})\) m on a horizontal football field. The motion of the ball is modelled as that of a particle moving horizontally with constant velocity \((5\mathbf{i} + 8\mathbf{j}) \text{ m s}^{-1}\). Find

1. the speed of the ball, [2]
2. the position vector of the ball after \(t\) seconds. [2]

The point \(B\) on the field has position vector \((10\mathbf{i} + 7\mathbf{j})\) m.

1. Find the time when the ball is due north of \(B\). [2]

At time \(t = 0\), another player starts running due north from \(B\) and moves with constant speed \(v \text{ m s}^{-1}\). Given that he intercepts the ball,

1. find the value of \(v\). [6]
2. State one physical factor, other than air resistance, which would be needed in a refinement of the model of the ball's motion to make the model more realistic. [1]

[In this question \(\mathbf{i}\) and \(\mathbf{j}\) are horizontal unit vectors due east and due north respectively.] A hiker \(H\) is walking with constant velocity \((1.2\mathbf{i} - 0.9\mathbf{j})\) m s\(^{-1}\).

1. Find the speed of \(H\). [2]

\includegraphics{figure_3} A horizontal field \(OABC\) is rectangular with \(OA\) due east and \(OC\) due north, as shown in Figure 3. At twelve noon hiker \(H\) is at the point \(Y\) with position vector \(100\mathbf{j}\) m, relative to the fixed origin \(O\).

1. Write down the position vector of \(H\) at time \(t\) seconds after noon. [2]

At noon, another hiker \(K\) is at the point with position vector \((9\mathbf{i} + 46\mathbf{j})\) m. Hiker \(K\) is moving with constant velocity \((0.75\mathbf{i} + 1.8\mathbf{j})\) m s\(^{-1}\).

1. Show that, at time \(t\) seconds after noon, $$\overrightarrow{HK} = [(9 - 0.45t)\mathbf{i} + (2.7t - 54)\mathbf{j}] \text{ metres.}$$ [4]

Hence,

1. show that the two hikers meet and find the position vector of the point where they meet. [5]

[In this question, the horizontal unit vectors \(\mathbf{i}\) and \(\mathbf{j}\) are directed due East and North respectively.] A coastguard station \(O\) monitors the movements of ships in a channel. At noon, the station's radar records two ships moving with constant speed. Ship \(A\) is at the point with position vector \((-5\mathbf{i} + 10\mathbf{j})\) km relative to \(O\) and has velocity \((2\mathbf{i} + 2\mathbf{j})\) km h\(^{-1}\). Ship \(B\) is at the point with position vector \((3\mathbf{i} + 4\mathbf{j})\) km and has velocity \((-2\mathbf{i} + 5\mathbf{j})\) km h\(^{-1}\).

1. Given that the two ships maintain these velocities, show that they collide. [6]

The coast guard radios ship \(A\) and orders it to reduce its speed to move with velocity \((\mathbf{i} + \mathbf{j})\) km h\(^{-1}\). Given that \(A\) obeys this order and maintains this new constant velocity,

1. find an expression for the vector \(\overrightarrow{AB}\) at time \(t\) hours after noon. [2]
2. find, to 3 significant figures, the distance between \(A\) and \(B\) at 1400 hours, [3]
3. Find the time at which \(B\) will be due north of \(A\). [2]