1.10f Distance between points: using position vectors

8 A pipeline is to be drilled under a river (see Fig. 8). With respect to axes Oxyz, with the \(x\)-axis pointing East, the \(y\)-axis North and the \(z\)-axis vertical, the pipeline is to consist of a straight section AB from the point \(\mathrm { A } ( 0 , - 40,0 )\) to the point \(\mathrm { B } ( 40,0 , - 20 )\) directly under the river, and another straight section BC . All lengths are in metres. \begin{figure}[h] \end{figure}

1. Calculate the distance AB . The section BC is to be drilled in the direction of the vector \(3 \mathbf { i } + 4 \mathbf { j } + \mathbf { k }\).
2. Find the angle ABC between the sections AB and BC . The section BC reaches ground level at the point \(\mathrm { C } ( a , b , 0 )\).
3. Write down a vector equation of the line BC . Hence find \(a\) and \(b\).
4. Show that the vector \(6 \mathbf { i } - 5 \mathbf { j } + 2 \mathbf { k }\) is perpendicular to the plane ABC . Hence find the cartesian equation of this plane.

7. Relative to a fixed origin, the points \(A\) and \(B\) have position vectors \(\left( \begin{array} { c } - 4 \\ 1 \\ 3 \end{array} \right)\) and \(\left( \begin{array} { c } - 3 \\ 6 \\ 1 \end{array} \right)\) respectively.

1. Find a vector equation for the line \(l _ { 1 }\) which passes through \(A\) and \(B\). The line \(l _ { 2 }\) has vector equation $$\mathbf { r } = \left( \begin{array} { c } 3 \\ - 7 \\ 9 \end{array} \right) + t \left( \begin{array} { c } 2 \\ - 3 \\ 1 \end{array} \right)$$
2. Show that lines \(l _ { 1 }\) and \(l _ { 2 }\) do not intersect.
3. Find the position vector of the point \(C\) on \(l _ { 2 }\) such that \(\angle A B C = 90 ^ { \circ }\).

8. The line \(l _ { 1 }\) passes through the points \(A\) and \(B\) with position vectors \(( - 3 \mathbf { i } + 3 \mathbf { j } + 2 \mathbf { k } )\) and ( \(7 \mathbf { i } - \mathbf { j } + 12 \mathbf { k }\) ) respectively, relative to a fixed origin.

1. Find a vector equation for \(l _ { 1 }\). The line \(l _ { 2 }\) has the equation $$\mathbf { r } = ( 5 \mathbf { j } - 7 \mathbf { k } ) + \mu ( \mathbf { i } - 2 \mathbf { j } + 7 \mathbf { k } )$$ The point \(C\) lies on \(l _ { 2 }\) and is such that \(A C\) is perpendicular to \(B C\).
2. Show that one possible position vector for \(C\) is \(( \mathbf { i } + 3 \mathbf { j } )\) and find the other. Assuming that \(C\) has position vector \(( \mathbf { i } + 3 \mathbf { j } )\),
3. find the area of triangle \(A B C\), giving your answer in the form \(k \sqrt { 5 }\).

7. The line \(l _ { 1 }\) passes through the points \(A\) and \(B\) with position vectors ( \(3 \mathbf { i } + 6 \mathbf { j } - 8 \mathbf { k }\) ) and ( \(8 \mathbf { j } - 6 \mathbf { k }\) ) respectively, relative to a fixed origin.

1. Find a vector equation for \(l _ { 1 }\). The line \(l _ { 2 }\) has vector equation $$\mathbf { r } = ( - 2 \mathbf { i } + 10 \mathbf { j } + 6 \mathbf { k } ) + \mu ( 7 \mathbf { i } - 4 \mathbf { j } + 6 \mathbf { k } ) ,$$ where \(\mu\) is a scalar parameter.
2. Show that lines \(l _ { 1 }\) and \(l _ { 2 }\) intersect.
3. Find the coordinates of the point where \(l _ { 1 }\) and \(l _ { 2 }\) intersect. The point \(C\) lies on \(l _ { 2 }\) and is such that \(A C\) is perpendicular to \(A B\).
4. Find the position vector of \(C\).

6. 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.The point \(A\) has position vector \(7 \mathbf { i } + 2 \mathbf { j } - 7 \mathbf { k }\) and the point \(B\) has position vector \(12 \mathbf { i } + 3 \mathbf { j } - 15 \mathbf { k }\) .

1. Find a vector for the line \(L _ { 1 }\) which passes through \(A\) and \(B\) . The line \(L _ { 2 }\) has vector equation $$\mathbf { r } = - 4 \mathbf { i } + 12 \mathbf { k } + \mu ( \mathbf { i } - 3 \mathbf { k } )$$
2. Show that \(L _ { 2 }\) passes through the origin \(O\) .
3. Show that \(L _ { 1 }\) and \(L _ { 2 }\) intersect at a point \(C\) and find the position vector of \(C\) .
4. Find the cosine of \(\angle O C A\) .
5. Hence,or otherwise,find the shortest distance from \(O\) to \(L _ { 1 }\) .
6. Show that \(| \overrightarrow { C O } | = | \overrightarrow { A B } |\) .
7. Find a vector equation for the line which bisects \(\angle O C A\) . \includegraphics[max width=\textwidth, alt={}, center]{f9d3e02c-cef2-435b-9cda-76c43fcac575-4_922_1054_279_586} Figure 1 shows a sketch of part of the curve with equation \(y = \mathrm { f } ( x )\), where $$\mathrm { f } ( x ) = x \left( 12 - x ^ { 2 } \right) .$$ The curve cuts the \(x\)-axis at the points \(P , O\) and \(R\), and \(Q\) is the maximum point.

7.The points \(O , P\) and \(Q\) lie on a circle \(C\) with diameter \(O Q\) .The position vectors of \(P\) and \(Q\) , relative to \(O\) ,are \(\mathbf { p }\) and \(\mathbf { q }\) respectively.

1. Prove that \(\mathbf { p } . \mathbf { q } = | \mathbf { p } | ^ { 2 }\) . \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{f2290882-b9a4-43ec-a38f-c44d46477242-6_615_714_412_689} \captionsetup{labelformat=empty} \caption{Figure 3}
   \end{figure} The point \(R\) also lies on \(C\) and \(O P Q R\) is a kite \(K\) as shown in Figure 3.The point \(S\) has position vector,relative to \(O\) ,of \(\lambda \mathbf { q }\) ,where \(\lambda\) is a constant.Given that \(\mathbf { p } = \mathbf { i } + 2 \mathbf { j } - \mathbf { k } , \mathbf { q } = 2 \mathbf { i } + \mathbf { j } - 2 \mathbf { k }\) and that \(O Q\) is perpendicular to \(P S\) ,find
2. the value of \(\lambda\) ,
3. the position vector of \(R\) ,
4. the area of \(K\) . Another circle \(C _ { 1 }\) is drawn inside \(K\) so that the 4 sides of the kite are each tangents to \(C _ { 1 }\) .
5. Find the radius of \(C _ { 1 }\) giving your answer in the form \(( \sqrt { } 2 - 1 ) \sqrt { } n\) ,where \(n\) is an integer. A second kite \(K _ { 1 }\) is similar to \(K\) and is drawn inside \(C _ { 1 }\) .
6. Find that area of \(K _ { 1 }\) .

7.Relative to a fixed origin \(O\) the points \(A , B\) and \(C\) have position vectors $$\mathbf { a } = - \mathbf { i } + \frac { 4 } { 3 } \mathbf { j } + 7 \mathbf { k } , \quad \mathbf { b } = 4 \mathbf { i } + \frac { 4 } { 3 } \mathbf { j } + 2 \mathbf { k } \text { and } \mathbf { c } = 6 \mathbf { i } + \frac { 16 } { 3 } \mathbf { j } + 2 \mathbf { k } \text { respectively. }$$

1. Find the cosine of angle \(A B C\) . The quadrilateral \(A B C D\) is a kite \(K\) .
2. Find the area of \(K\) . A circle is drawn inside \(K\) so that it touches each of the 4 sides of \(K\) .
3. Find the radius of the circle,giving your answer in the form \(p \sqrt { } ( q ) - q \sqrt { } ( p )\) ,where \(p\) and \(q\) are positive integers.
4. Find the position vector of the point \(D\) .
   (Total 18 marks)

3.The points \(A , B , C , D\) and \(E\) are five of the vertices of a rectangular cuboid and \(A E\) is a diagonal of the cuboid.With respect to a fixed origin \(O\) ,the position vectors of \(A , B , C\) and \(D\) are \(\mathbf { a , b , c }\) and \(\mathbf{d}\) respectively,where $$\mathbf { a } = \left( \begin{array} { c } 1 \\ 2 \\ - 1 \end{array} \right) , \quad \mathbf { b } = \left( \begin{array} { c } 0 \\ - 3 \\ - 8 \end{array} \right) , \quad \mathbf { c } = \left( \begin{array} { c } 4 \\ - 1 \\ - 10 \end{array} \right)$$

\text { and } \mathbf { d } = \left( \begin{array} { c } - 4
2
- 11 \end{array} \right)$$

1. Find the position vector of \(E\) . The volume of a tetrahedron is given by the formula $$\text { volume } = \frac { 1 } { 3 } ( \text { area of base } ) \times ( \text { height } )$$
2. Find the volume of the tetrahedron \(A B C D\) . 4.(a)Given that \(x > 0 , y > 0 , x \neq 1\) and \(n > 0\) ,show that $$\log _ { x } y = \log _ { x ^ { n } } y ^ { n }$$
3. Solve the following,leaving your answers in the form \(2 ^ { p }\) ,where \(p\) is a rational number.
   1. \(\log _ { 2 } u + \log _ { 4 } u ^ { 2 } + \log _ { 8 } u ^ { 3 } + \log _ { 16 } u ^ { 4 } = 5\)
   2. \(\log _ { 2 } v + \log _ { 4 } v + \log _ { 8 } v + \log _ { 16 } v = 5\)
   3. \(\log _ { 4 } w ^ { 2 } + \frac { 3 \log _ { 8 } 64 } { \log _ { 2 } w } = 5\)

11. With respect to a fixed origin \(O\), the lines \(l _ { 1 }\) and \(l _ { 2 }\) are given by the equations $$\begin{aligned} & l _ { 1 } : \mathbf { r } = \left( \begin{array} { l } 7 \\ 4 \\ 9 \end{array} \right) + \lambda \left( \begin{array} { l } 1 \\ 1 \\ 4 \end{array} \right) \\ & l _ { 2 } : \mathbf { r } = \left( \begin{array} { r } - 6 \\ - 7 \\ 3 \end{array} \right) + \mu \left( \begin{array} { l } 5 \\ 4 \\ b \end{array} \right) \end{aligned}$$ where \(\lambda\) and \(\mu\) are scalar parameters and \(b\) is a constant.
Given that \(l _ { 1 }\) and \(l _ { 2 }\) meet at the point \(X\),

1. show that \(b = - 3\) and find the coordinates of \(X\). The point \(A\) lies on \(l _ { 1 }\) and has coordinates (6, 3, 5)
   The point \(B\) lies on \(l _ { 2 }\) and has coordinates \(( 14,9 , - 9 )\)
2. Show that angle \(A X B = \arccos \left( - \frac { 1 } { 10 } \right)\)
3. Using the result obtained in part (b), find the exact area of triangle \(A X B\). Write your answer in the form \(p \sqrt { q }\) where \(p\) and \(q\) are integers to be determined.

7 The line with equation \(3 x + 4 y - 10 = 0\) passes through point \(A ( 2,1 )\) and point \(B ( 10 , k )\).

1. Find the value of \(k\).
2. Calculate the length of \(A B\). A circle has equation \(( x - 6 ) ^ { 2 } + ( y + 2 ) ^ { 2 } = 25\).
3. Write down the coordinates of the centre and the radius of the circle.
4. Verify that \(A B\) is a diameter of the circle.

1 The points \(A\) and \(B\) have coordinates \(( 6,1 )\) and \(( - 2,7 )\) respectively.

1. Find the length of \(A B\).
2. Find the gradient of the line \(A B\).
3. Determine whether the line \(4 x - 3 y - 10 = 0\) is perpendicular to \(A B\).

7 The equation of a straight line \(l\) is $$\mathbf { r } = \left( \begin{array} { l } 1 \\ 0 \\ 2 \end{array} \right) + t \left( \begin{array} { r } 1 \\ - 1 \\ 0 \end{array} \right) .$$ \(O\) is the origin.

1. Find the position vector of the point \(P\) on \(l\) such that \(O P\) is perpendicular to \(l\).
2. A point \(Q\) on \(l\) is such that the length of \(O Q\) is 3 units. Find the two possible position vectors of \(Q\). [3]

10 Lines \(l _ { 1 }\) and \(l _ { 2 }\) have vector equations $$\mathbf { r } = - \mathbf { i } + 2 \mathbf { j } + 7 \mathbf { k } + t ( 2 \mathbf { i } + 2 \mathbf { j } + \mathbf { k } ) \quad \text { and } \quad \mathbf { r } = 2 \mathbf { i } + 9 \mathbf { j } - 4 \mathbf { k } + s ( \mathbf { i } + 3 \mathbf { j } - 2 \mathbf { k } )$$ respectively. The point \(A\) has coordinates ( \(- 3,0,6\) ) relative to the origin \(O\).

1. Show that \(A\) lies on \(l _ { 1 }\) and that \(O A\) is perpendicular to \(l _ { 1 }\).
2. Show that the line through \(O\) and \(A\) intersects \(l _ { 2 }\).
3. Given that the point of intersection in part (ii) is \(B\), find the ratio \(| \overrightarrow { O A } | : | \overrightarrow { B A } |\). \section*{THERE ARE NO QUESTIONS WRITTEN ON THIS PAGE.}

7 Points \(A ( 2,2,5 ) , B ( 1 , - 1 , - 4 ) , C ( 3,3,10 )\) and \(D ( 8,6,3 )\) are the vertices of a pyramid with a triangular base.

1. Calculate the lengths \(A B\) and \(A C\), and the angle \(B A C\).
2. Show that \(\overrightarrow { A D }\) is perpendicular to both \(\overrightarrow { A B }\) and \(\overrightarrow { A C }\).
3. Calculate the volume of the pyramid \(A B C D\).
   [0pt] [The volume of the pyramid is \(V = \frac { 1 } { 3 } \times\) base area × perpendicular height.]

7 A tent has vertices ABCDEF with coordinates as shown in Fig. 7. Lengths are in metres. The \(\mathrm { O } x y\) plane is horizontal. \begin{figure}[h] \end{figure}

1. Find the length of the ridge of the tent DE , and the angle this makes with the horizontal.
2. Show that the vector \(\mathbf { i } - 4 \mathbf { j } + 5 \mathbf { k }\) is normal to the plane through \(\mathrm { A } , \mathrm { D }\) and E . Hence find the equation of this plane. Given that B lies in this plane, find \(a\).
3. Verify that the equation of the plane BCD is \(x + z = 8\). Hence find the acute angle between the planes ABDE and BCD .

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.

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 \(O A : A R\).

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.

2 The points \(A\) and \(B\) have position vectors \(\left( \begin{array} { c } 1 \\ - 2 \\ 5 \end{array} \right)\) and \(\left( \begin{array} { c } - 3 \\ - 1 \\ 2 \end{array} \right)\) respectively.

1. Find the exact length of \(A B\).
2. Find the position vector of the midpoint of \(A B\). The points \(P\) and \(Q\) have position vectors \(\left( \begin{array} { l } 1 \\ 2 \\ 0 \end{array} \right)\) and \(\left( \begin{array} { l } 5 \\ 1 \\ 3 \end{array} \right)\) respectively.
3. Show that \(A B P Q\) is a parallelogram.

9 Points \(A , B\) and \(C\) have position vectors \(\mathbf { a } , \mathbf { b }\) and \(\mathbf { c }\) relative to an origin \(O\) in 3-dimensional space. Rectangles \(O A D C\) and \(B E F G\) are the base and top surface of a cuboid. \includegraphics[max width=\textwidth, alt={}, center]{7298e7b9-ad52-480c-bc2b-8289aeab9ebb-07_522_812_952_280}

• The point \(M\) is the midpoint of \(B C\).
• The point \(X\) lies on \(A M\) such that \(A X = 2 X M\).
  1. Find \(\overrightarrow { O X }\) in terms of \(\mathbf { a } , \mathbf { b }\) and \(\mathbf { c }\), simplifying your answer.
  2. Hence show that the lines \(O F\) and \(A M\) intersect.

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

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. \hspace{0pt} [In this question the unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are due east and due north respectively.]

A stone slides horizontally across ice.
Initially the stone is at the point \(A ( - 24 \mathbf { i } - 10 \mathbf { j } ) \mathrm { m }\) relative to a fixed point \(O\).
After 4 seconds the stone is at the point \(B ( 12 \mathbf { i } + 5 \mathbf { j } )\) m relative to the fixed point \(O\).
The motion of the stone is modelled as that of a particle moving in a straight line at constant speed. Using the model,

1. prove that the stone passes through \(O\),
2. calculate the speed of the stone.