1.10c Magnitude and direction: of vectors

3 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.

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.

The points \(A , B\) and \(C\) have position vectors \(\mathbf { i } , 2 \mathbf { j }\) and \(4 \mathbf { k }\) respectively, relative to an origin \(O\). The point \(N\) is the foot of the perpendicular from \(O\) to the plane \(A B C\). The point \(P\) on the line-segment \(O N\) is such that \(O P = \frac { 3 } { 4 } O N\). The line \(A P\) meets the plane \(O B C\) at \(Q\).

1. Find a vector perpendicular to the plane \(A B C\) and show that the length of \(O N\) is \(\frac { 4 } { \sqrt { } ( 21 ) }\).
2. Find the position vector of the point \(Q\).
3. Show that the acute angle between the planes \(A B C\) and \(A B Q\) is \(\cos ^ { - 1 } \left( \frac { 2 } { 3 } \right)\).

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)

6. [In this question, \(\mathbf { i }\) is a horizontal unit vector and \(\mathbf { j }\) is an upward vertical unit vector.] A particle \(P\) is projected from a fixed origin \(O\) with velocity ( \(3 \mathbf { i } + 4 \mathbf { j }\) ) \(\mathrm { m } \mathrm { s } ^ { - 1 }\). The particle moves freely under gravity and passes through the point \(A\) with position vector \(\lambda ( \mathbf { i } - \mathbf { j } ) \mathrm { m }\), where \(\lambda\) is a positive constant.

1. Find the value of \(\lambda\).
2. Find
   1. the speed of \(P\) at the instant when it passes through \(A\),
   2. the direction of motion of \(P\) at the instant when it passes through \(A\).
      HMAV SIHI NITIIIUM ION OC
      VILV SIHI NI JAHM ION OC
      VJ4V SIHI NI JIIYM ION OC

8

1. A curve has cartesian equation \(x y = 8\). Show that the polar equation of the curve is \(r ^ { 2 } = 16 \operatorname { cosec } 2 \theta\).
2. The diagram shows a sketch of the curve, \(C\), whose polar equation is $$r ^ { 2 } = 16 \operatorname { cosec } 2 \theta , \quad 0 < \theta < \frac { \pi } { 2 }$$ \includegraphics[max width=\textwidth, alt={}, center]{c4bce668-61f1-4be0-97ee-c635df7e1fc6-4_364_567_1635_726}
   1. Find the polar coordinates of the point \(N\) which lies on the curve \(C\) and is closest to the pole \(O\).
   2. The circle whose polar equation is \(r = 4 \sqrt { 2 }\) intersects the curve \(C\) at the points \(P\) and \(Q\). Find, in an exact form, the polar coordinates of \(P\) and \(Q\).
   3. The obtuse angle \(P N Q\) is \(\alpha\) radians. Find the value of \(\alpha\), giving your answer to three significant figures.
      (5 marks)

2 The point \(A\) is such that the magnitude of \(\overrightarrow { O A }\) is 8 and the direction of \(\overrightarrow { O A }\) is \(240 ^ { \circ }\).

1. 1. Show the point \(A\) on the axes provided in the Printed Answer Booklet.
   2. Find the position vector of point \(A\). Give your answer in terms of \(\mathbf { i }\) and \(\mathbf { j }\). The point \(B\) has position vector \(6 \mathbf { i }\).
2. Find the exact area of triangle \(A O B\). The point \(C\) is such that \(O A B C\) is a parallelogram.
3. Find the position vector of \(C\). Give your answer in terms of \(\mathbf { i }\) and \(\mathbf { j }\).

4 It is given that \(A B C D\) is a quadrilateral. The position vector of \(A\) is \(\mathbf { i } + \mathbf { j }\), and the position vector of \(B\) is \(3 \mathbf { i } + 5 \mathbf { j }\).

1. Find the length \(A B\).
2. The position vector of \(C\) is \(p \mathbf { i } + p \mathbf { j }\) where \(p\) is a constant greater than 1 . Given that the length \(A B\) is equal to the length \(B C\), determine the position vector of \(C\).
3. The point \(M\) is the midpoint of \(A C\). Given that \(\overrightarrow { M D } = 2 \overrightarrow { B M }\), determine the position vector of \(D\).
4. State the name of the quadrilateral \(A B C D\), giving a reason for your answer.

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 \(P\) 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 \(P Q R S\)

2. Relative to a fixed origin \(O\),
the point \(A\) has position vector \(( 3 \mathbf { i } - \mathbf { j } + 2 \mathbf { k } )\) the point \(B\) has position vector ( \(\mathbf { i } + 2 \mathbf { j } - 4 \mathbf { k }\) )
and the point \(C\) has position vector \(( - \mathbf { i } + \mathbf { j } + a \mathbf { k } )\), where \(a\) is a constant and \(a > 0\).
Given that \(| \overrightarrow { B C } | = \sqrt { 41 }\) a. show that \(a = 2\). \(D\) is the point such that \(A B C D\) forms a parallelogram.
b. Find the position vector of \(D\).

9. \begin{figure}[h] \end{figure} Figure 3 shows a sketch of a parallelogram \(P Q R S\).
Given that

• \(\overrightarrow { P Q } = 2 \mathbf { i } + 3 \mathbf { j } - 4 \mathbf { k }\)
• \(\overrightarrow { Q R } = 5 \mathbf { i } - 2 \mathbf { k }\)
  1. show that parallelogram \(P Q R S\) is a rhombus.
  2. Find the exact area of the rhombus \(P Q R S\).

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

• the point \(A\) has position vector \(5 \mathbf { i } + 3 \mathbf { j } + 2 \mathbf { k }\)
• the point \(B\) has position vector \(2 \mathbf { i } + 4 \mathbf { j } + a \mathbf { k }\) where \(a\) is a positive integer.
  1. Show that \(| \overrightarrow { O A } | = \sqrt { 38 }\)
  2. Find the smallest value of \(a\) for which

$$| \overrightarrow { O B } | > | \overrightarrow { O A } |$$

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 \(a\).

1. At time \(t\) seconds, where \(t \geqslant 0\), a particle \(P\) has velocity \(\mathbf { v } \mathrm { ms } ^ { - 1 }\) where

$$\mathbf { v } = \left( t ^ { 2 } - 3 t + 7 \right) \mathbf { i } + \left( 2 t ^ { 2 } - 3 \right) \mathbf { j }$$ Find

1. the speed of \(P\) at time \(t = 0\)
2. the value of \(t\) when \(P\) is moving parallel to \(( \mathbf { i } + \mathbf { j } )\)
3. the acceleration of \(P\) at time \(t\) seconds
4. the value of \(t\) when the direction of the acceleration of \(P\) is perpendicular to \(\mathbf { i }\)

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 \(O\) ]

A particle \(P\) is moving on a smooth horizontal plane.
The particle has constant acceleration \(( 2.4 \mathbf { i } + \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 2 }\) At time \(t = 0 , P\) passes through the point \(A\).
At time \(t = 5 \mathrm {~s} , P\) passes through the point \(B\).
The velocity of \(P\) as it passes through \(A\) is \(( - 16 \mathbf { i } - 3 \mathbf { j } ) \mathrm { ms } ^ { - 1 }\)

1. Find the speed of \(P\) as it passes through \(B\). The position vector of \(A\) is \(( 44 \mathbf { i } - 10 \mathbf { j } ) \mathrm { m }\).
   At time \(t = T\) seconds, where \(T > 5 , P\) passes through the point \(C\).
   The position vector of \(C\) is \(( 4 \mathbf { i } + c \mathbf { j } ) \mathrm { m }\).
2. Find the value of \(T\).
3. Find the value of \(c\).

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

\section*{Solutions relying entirely on calculator technology are not acceptable.} [In this question, \(\mathbf { i }\) is a unit vector due east and \(\mathbf { j }\) is a unit vector due north.
Position vectors are given relative to a fixed origin \(O\).] At time \(t\) seconds, \(t \geqslant 1\), the position vector of a particle \(P\) is \(\mathbf { r }\) metres, where $$\mathbf { r } = c t ^ { \frac { 1 } { 2 } } \mathbf { i } - \frac { 3 } { 8 } t ^ { 2 } \mathbf { j }$$ and \(c\) is a constant.
When \(t = 4\), the bearing of \(P\) from \(O\) is \(135 ^ { \circ }\)

1. Show that \(c = 3\)
2. Find the speed of \(P\) when \(t = 4\) When \(t = T , P\) is accelerating in the direction of ( \(\mathbf { - i } - \mathbf { 2 7 j }\) ).
3. Find the value of \(T\).

2 Points \(A\) and \(B\) have position vectors \(\binom { - 3 } { 4 }\) and \(\binom { 1 } { 2 }\) respectively.
Point \(C\) has position vector \(\binom { p } { 1 }\) and \(A B C\) is a straight line.

1. Find \(p\). Point \(D\) has position vector \(\binom { q } { 1 }\) and angle \(A B D = 90 ^ { \circ }\).
2. Determine the value of \(q\).

5 Points \(A , B , C\) and \(D\) have position vectors \(\mathbf { a } = \binom { 1 } { 2 } , \mathbf { b } = \binom { 3 } { 5 } , \mathbf { c } = \binom { 7 } { 4 }\) and \(\mathbf { d } = \binom { 4 } { k }\).

1. Find the value of \(k\) for which \(D\) is the midpoint of \(A C\).
2. Find the two values of \(k\) for which \(| \overrightarrow { A D } | = \sqrt { 13 }\).
3. Find one value of \(k\) for which the four points form a trapezium.

7 \includegraphics[max width=\textwidth, alt={}, center]{a1f4ccbd-f5ed-437a-ae76-c4925ce86e25-06_648_586_255_244} The diagram shows the parallelogram \(O A C B\) where \(\overrightarrow { O A } = 2 \mathbf { i } + 4 \mathbf { j }\) and \(\overrightarrow { O B } = 4 \mathbf { i } - 3 \mathbf { j }\).

1. Show that \(\cos A O B = - \frac { 2 \sqrt { 5 } } { 25 }\).
2. Hence find the exact value of \(\sin A O B\).
3. Determine the area of \(O A C B\).

6 The vertices of triangle \(A B C\) are \(A ( - 3,1 ) , B ( 5,0 )\) and \(C ( 9,7 )\).

1. Show that \(A B = B C\).
2. Show that angle \(A B C\) is not a right angle.
3. Find the coordinates of the midpoint of \(A C\).
4. Determine the equation of the line of symmetry of the triangle, giving your answer in the form \(p x + q y = r\), where \(p , q\) and \(r\) are integers to be determined.
5. Write down an equation of the circle with centre \(A\) which passes through \(B\). This circle intersects the line of symmetry of the triangle at \(B\) and at a second point.
6. Find the coordinates of this second point.