1.10d Vector operations: addition and scalar multiplication

4 The points A , B and C have coordinates \(( 2,0 , - 1 ) , ( 4,3 , - 6 )\) and \(( 9,3 , - 4 )\) respectively.

1. Show that AB is perpendicular to BC .
2. Find the area of triangle ABC .

6 Fig. 6 shows a lean-to greenhouse ABCDHEFG . With respect to coordinate axes Oxyz , the coordinates of the vertices are as shown. All distances are in metres. Ground level is the plane \(z = 0\). \begin{figure}[h] \end{figure}

1. Verify that the equation of the plane through \(\mathrm { A } , \mathrm { B }\) and E is \(x + 6 y + 12 = 0\). Hence, given that F lies in this plane, show that \(a = - 2 \frac { 1 } { 3 }\).
2. (A) Show that the vector \(\left( \begin{array} { r } 1 \\ - 6 \\ 0 \end{array} \right)\) is normal to the plane DHC.
   (B) Hence find the cartesian equation of this plane.
   (C) Given that G lies in the plane DHC , find \(b\) and the length FG .
3. Find the angle EFB . A straight wire joins point H to a point P which is half way between E and F . Q is a point two-thirds of the way along this wire, so that \(\mathrm { HQ } = 2 \mathrm { QP }\).
4. Find the height of Q above the ground. \section*{Question 7 begins on page 4.}

3 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 \(\binom { 2 } { 3 } \mathrm {~ms} ^ { - 2 }\).

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

3 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.

8 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.

With respect to an origin \(O\), the point \(A\) has position vector \(4 \mathbf { i } - 2 \mathbf { j } + 2 \mathbf { k }\) and the plane \(\Pi _ { 1 }\) has equation $$\mathbf { r } = ( 4 + \lambda + 3 \mu ) \mathbf { i } + ( - 2 + 7 \lambda + \mu ) \mathbf { j } + ( 2 + \lambda - \mu ) \mathbf { k } ,$$ where \(\lambda\) and \(\mu\) are real. The point \(L\) is such that \(\overrightarrow { O L } = 3 \overrightarrow { O A }\) and \(\Pi _ { 2 }\) is the plane through \(L\) which is parallel to \(\Pi _ { 1 }\). The point \(M\) is such that \(\overrightarrow { A M } = 3 \overrightarrow { M L }\).

1. Show that \(A\) is in \(\Pi _ { 1 }\).
2. Find a vector perpendicular to \(\Pi _ { 2 }\).
3. Find the position vector of the point \(N\) in \(\Pi _ { 2 }\) such that \(O N\) is perpendicular to \(\Pi _ { 2 }\).
4. Show that the position vector of \(M\) is \(10 \mathbf { i } - 5 \mathbf { j } + 5 \mathbf { k }\) and find the perpendicular distance of \(M\) from the line through \(O\) and \(N\), giving your answer correct to 3 significant figures.

3. A particle of mass 0.6 kg is moving with constant velocity ( \(c \mathbf { i } + 2 c \mathbf { j }\) ) \(\mathrm { ms } ^ { - 1 }\), where \(c\) is a positive constant. The particle receives an impulse of magnitude \(2 \sqrt { 10 } \mathrm {~N} \mathrm {~s}\). Immediately after receiving the impulse the particle has velocity ( \(2 c \mathbf { i } - c \mathbf { j }\) ) \(\mathrm { ms } ^ { - 1 }\). Find the value of \(c\).
(6)

4. A particle \(P\) moves in a straight line with constant velocity. Initially \(P\) is at the point \(A\) with position vector \(( 2 \mathbf { i } - \mathbf { j } ) \mathrm { m }\) relative to a fixed origin \(O\), and 2 s later it is at the point \(B\) with position vector \(( 6 \mathbf { i } + \mathbf { j } ) \mathrm { m }\).

1. Find the velocity of \(P\).
2. Find, in degrees to one decimal place, the size of the angle between the direction of motion of \(P\) and the vector \(\mathbf { i }\). Three seconds after it passes \(B\) the particle \(P\) reaches the point \(C\).
3. Find, in m to one decimal place, the distance \(O C\).

5 \includegraphics[max width=\textwidth, alt={}, center]{febe231d-200a-4957-b41b-de5b9be98b0a-5_424_583_255_244} The diagram shows points \(A\) and \(B\), which have position vectors \(\mathbf { a }\) and \(\mathbf { b }\) with respect to an origin \(O\). \(P\) is the point on \(O B\) such that \(O P : P B = 3 : 1\) and \(Q\) is the midpoint of \(A B\).

1. Find \(\overrightarrow { P Q }\) in terms of \(\mathbf { a }\) and \(\mathbf { b }\). The line \(O A\) is extended to a point \(R\), so that \(P Q R\) is a straight line.
2. Explain why \(\overrightarrow { P R } = k ( 2 \mathbf { a } - \mathbf { b } )\), where \(k\) is a constant.
3. Hence determine the ratio \(O A : A R\).

1. The line \(l _ { 1 }\) has equation \(4 y - 3 x = 10\)

The line \(l _ { 2 }\) passes through the points \(( 5 , - 1 )\) and \(( - 1,8 )\).
Determine, giving full reasons for your answer, whether lines \(l _ { 1 }\) and \(l _ { 2 }\) are parallel, perpendicular or neither.

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 \(\mathrm { kmh } ^ { - 1 }\)

The triangle \(P Q R\) is such that \(\overrightarrow { P Q } = 3 \mathbf { i } + 5 \mathbf { j }\) and \(\overrightarrow { P R } = 13 \mathbf { i } - 15 \mathbf { j }\)

1. Find \(\overrightarrow { Q R }\)
2. Hence find \(| \overrightarrow { Q R } |\) giving your answer as a simplified surd. The point \(S\) lies on the line segment \(Q R\) so that \(Q S : S R = 3 : 2\)
3. Find \(\overrightarrow { P S }\)

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

• point \(A\) has position vector \(10 \mathbf { i } - 3 \mathbf { j }\)
• point \(B\) has position vector \(- 8 \mathbf { i } + 9 \mathbf { j }\)
• point \(C\) has position vector \(- 2 \mathbf { i } + p \mathbf { j }\) where \(p\) is a constant
  1. Find \(\overrightarrow { A B }\)
  2. Find \(| \overrightarrow { A B } |\) giving your answer as a fully simplified surd.

Given that points \(A , B\) and \(C\) lie on a straight line,

1. find the value of \(p\),
2. state the ratio of the area of triangle \(A O C\) to the area of triangle \(A O B\).

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

• point \(P\) has position vector \(9 \mathbf { i } - 8 \mathbf { j }\)
• point \(Q\) has position vector \(3 \mathbf { i } - 5 \mathbf { j }\)
  1. Find \(\overrightarrow { P Q }\)

Given that \(R\) is the point such that \(\overrightarrow { Q R } = 9 \mathbf { i } + 18 \mathbf { j }\)

show that angle \(P Q R = 90 ^ { \circ }\) Given also that \(S\) is the point such that \(\overrightarrow { P S } = 3 \overrightarrow { Q R }\)

find the exact area of \(P Q R S\)

1. \hspace{0pt} [In this question, \(\mathbf { i }\) and \(\mathbf { j }\) are perpendi cular unit vectors in a horizontal plane]

A particle P is moving on a smooth horizontal surface under the action of two forces.
Given that

• the mass of P is 2 kg
• the two forces are \(( 2 \mathbf { i } + 4 \mathbf { j } ) \mathrm { N }\) and \(( \mathbf { i } - 2 \mathbf { j } ) \mathrm { N }\), where C is a constant
• the magnitude of the acceleration of P is \(\sqrt { 5 } \mathrm {~m} \mathrm {~s} ^ { - 2 }\) find the two possible values of C .

3. Relative to a fixed origin,

• point \(A\) has position vector \(- 2 \mathbf { i } + 4 \mathbf { j } + 7 \mathbf { k }\)
• point \(B\) has position vector \(- \mathbf { i } + 3 \mathbf { j } + 8 \mathbf { k }\)
• point \(C\) has position vector \(\mathbf { i } + \mathbf { j } + 4 \mathbf { k }\)
• point \(D\) has position vector \(- \mathbf { i } + 3 \mathbf { j } + 2 \mathbf { k }\) a. Show that \(\overrightarrow { A B }\) and \(\overrightarrow { C D }\) are parallel and the ratio \(\overrightarrow { A B } : \overrightarrow { C D }\) in its simplest form.
  b. Hence describe the quadrilateral \(A B C D\).

9. \begin{figure}[h] \end{figure} Figure 3 shows a sketch of a parallelogram \(X A P B\).
Given that \(\overrightarrow { O X } = \left( \begin{array} { l } 1 \\ 2 \\ 3 \end{array} \right)\) $$\begin{aligned} & \overrightarrow { O A } = \left( \begin{array} { l } 0 \\ 4 \\ 1 \end{array} \right) \\ & \overrightarrow { O B } = \left( \begin{array} { l } 3 \\ 3 \\ 1 \end{array} \right) \end{aligned}$$ a. Find the coordinates of the point \(P\).
b. Show that \(X A P B\) is a rhombus.
c. Find the exact area of the rhombus \(X A P B\).

2. \includegraphics[max width=\textwidth, alt={}, center]{cb92f7b6-2ba5-4703-9595-9ba8570fc52b-04_656_725_283_635} \section*{Figure 1} Figure 1 shows a triangle \(O A C\) where \(O B\) divides \(A C\) in the ratio \(2 : 3\).
Show that \(\mathbf { b } = \frac { 1 } { 5 } ( 3 \mathbf { a } + 2 \mathbf { c } )\)

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

• point \(A\) has position vector \(2 \mathbf { i } + 5 \mathbf { j } - 6 \mathbf { k }\)
• point \(B\) has position vector \(3 \mathbf { i } - 3 \mathbf { j } - 4 \mathbf { k }\)
• point \(C\) has position vector \(2 \mathbf { i } - 16 \mathbf { j } + 4 \mathbf { k }\)
  1. Find \(\overrightarrow { A B }\)
  2. Show that quadrilateral \(O A B C\) is a trapezium, giving reasons for your answer.

1. In this question you should show all stages of your working.

\section*{Solutions relying entirely on calculator technology are not acceptable.} A company made a profit of \(\pounds 20000\) in its first year of trading, Year 1
A model for future trading predicts that the yearly profit will increase by \(8 \%\) each year, so that the yearly profits will form a geometric sequence. According to the model,

1. show that the profit for Year 3 will be \(\pounds 23328\)
2. find the first year when the yearly profit will exceed £65000
3. find the total profit for the first 20 years of trading, giving your answer to the nearest £1000

6. A company plans to extract oil from an oil field. The daily volume of oil \(V\), measured in barrels that the company will extract from this oil field depends upon the time, \(t\) years, after the start of drilling. The company decides to use a model to estimate the daily volume of oil that will be extracted. The model includes the following assumptions:

• The initial daily volume of oil extracted from the oil field will be 16000 barrels.
• The daily volume of oil that will be extracted exactly 4 years after the start of drilling will be 9000 barrels.
• The daily volume of oil extracted will decrease over time.

The diagram below shows the graphs of two possible models. \begin{figure}[h] \end{figure} \begin{figure}[h] \end{figure}

1. 1. Use model \(A\) to estimate the daily volume of oil that will be extracted exactly 3 years after the start of drilling.
   2. Write down a limitation of using model \(A\).
2. 1. Using an exponential model and the information given in the question, find a possible equation for model \(B\).
   2. Using your answer to (b)(i) estimate the daily volume of oil that will be extracted exactly 3 years after the start of drilling.

Relative to a fixed origin \(O\),
the point \(A\) has position vector \(\mathbf { i } + 7 \mathbf { j } - 2 \mathbf { k }\),
the point \(B\) has position vector \(4 \mathbf { i } + 3 \mathbf { j } + 3 \mathbf { k }\),
and the point \(C\) has position vector \(2 \mathbf { i } + 10 \mathbf { j } + 9 \mathbf { k }\).
Given that \(A B C D\) is a parallelogram,

1. find the position vector of point \(D\). The vector \(\overrightarrow { A X }\) has the same direction as \(\overrightarrow { A B }\).
   Given that \(| \overrightarrow { A X } | = 10 \sqrt { 2 }\),
2. find the position vector of \(X\).

Relative to a fixed origin \(O\),
the point \(A\) has position vector \(( 2 \mathbf { i } + 3 \mathbf { j } - 4 \mathbf { k } )\),
the point \(B\) has position vector ( \(4 \mathbf { i } - 2 \mathbf { j } + 3 \mathbf { k }\) ),
and the point \(C\) has position vector ( \(a \mathbf { i } + 5 \mathbf { j } - 2 \mathbf { k }\) ), where \(a\) is a constant and \(a < 0 D\) is the point such that \(\overrightarrow { A B } = \overrightarrow { B D }\).

1. Find the position vector of \(D\). Given \(| \overrightarrow { A C } | = 4\)
2. find the value of \(a\).

10. Figure 7 Figure 7 shows a sketch of triangle \(O A B\).
The point \(C\) is such that \(\overrightarrow { O C } = 2 \overrightarrow { O A }\).
The point \(M\) is the midpoint of \(A B\).
The straight line through \(C\) and \(M\) cuts \(O B\) at the point \(N\).
Given \(\overrightarrow { O A } = \mathbf { a }\) and \(\overrightarrow { O B } = \mathbf { b }\)

1. Find \(\overrightarrow { C M }\) in terms of \(\mathbf { a }\) and \(\mathbf { b }\)
2. Show that \(\overrightarrow { O N } = \left( 2 - \frac { 3 } { 2 } \lambda \right) \mathbf { a } + \frac { 1 } { 2 } \lambda \mathbf { b }\), where \(\lambda\) is a scalar constant.
3. Hence prove that \(O N : N B = 2 : 1\)