1.10f Distance between points: using position vectors

8. With respect to a fixed origin \(O\), the lines \(l _ { 1 }\) and \(l _ { 2 }\) are given by the equations $$l _ { 1 } : \mathbf { r } = \left( \begin{array} { r } 2 \\ - 3 \\ 4 \end{array} \right) + \lambda \left( \begin{array} { r } - 1 \\ 2 \\ 1 \end{array} \right) , \quad l _ { 2 } : \mathbf { r } = \left( \begin{array} { r } 2 \\ - 3 \\ 4 \end{array} \right) + \mu \left( \begin{array} { r } 5 \\ - 2 \\ 5 \end{array} \right)$$ where \(\lambda\) and \(\mu\) are scalar parameters.

1. Find, to the nearest \(0.1 ^ { \circ }\), the acute angle between \(l _ { 1 }\) and \(l _ { 2 }\) The point \(A\) has position vector \(\left( \begin{array} { l } 0 \\ 1 \\ 6 \end{array} \right)\).
2. Show that \(A\) lies on \(l _ { 1 }\) The lines \(l _ { 1 }\) and \(l _ { 2 }\) intersect at the point \(X\).
3. Write down the coordinates of \(X\).
4. Find the exact value of the distance \(A X\). The distinct points \(B _ { 1 }\) and \(B _ { 2 }\) both lie on the line \(l _ { 2 }\) Given that \(A X = X B _ { 1 } = X B _ { 2 }\)
5. find the area of the triangle \(A B _ { 1 } B _ { 2 }\) giving your answer to 3 significant figures. Given that the \(x\) coordinate of \(B _ { 1 }\) is positive,
6. find the exact coordinates of \(B _ { 1 }\) and the exact coordinates of \(B _ { 2 }\) \includegraphics[max width=\textwidth, alt={}, center]{245bbe52-3a14-4494-af17-7711caf79b22-28_96_59_2478_1834}

1. The line \(l _ { 1 }\) has vector equation

$$\mathbf { r } = \left( \begin{array} { l } 3 \\ 1 \\ 2 \end{array} \right) + \lambda \left( \begin{array} { r } 1 \\ - 1 \\ 4 \end{array} \right)$$ and the line \(l _ { 2 }\) has vector equation $$\mathbf { r } = \left( \begin{array} { r } 0 \\ 4 \\ - 2 \end{array} \right) + \mu \left( \begin{array} { r } 1 \\ - 1 \\ 0 \end{array} \right) ,$$ where \(\lambda\) and \(\mu\) are parameters.
The lines \(l _ { 1 }\) and \(l _ { 2 }\) intersect at the point \(B\) and the acute angle between \(l _ { 1 }\) and \(l _ { 2 }\) is \(\theta\).

1. Find the coordinates of \(B\).
2. Find the value of \(\cos \theta\), giving your answer as a simplified fraction. The point \(A\), which lies on \(l _ { 1 }\), has position vector \(\mathbf { a } = 3 \mathbf { i } + \mathbf { j } + 2 \mathbf { k }\).
   The point \(C\), which lies on \(l _ { 2 }\), has position vector \(\mathbf { c } = 5 \mathbf { i } - \mathbf { j } - 2 \mathbf { k }\).
   The point \(D\) is such that \(A B C D\) is a parallelogram.
3. Show that \(| \overrightarrow { A B } | = | \overrightarrow { B C } |\).
4. Find the position vector of the point \(D\).

1. The point \(A\), with coordinates \(( 0 , a , b )\) lies on the line \(l _ { 1 }\), which has equation

$$\mathbf { r } = 6 \mathbf { i } + 19 \mathbf { j } - \mathbf { k } + \lambda ( \mathbf { i } + 4 \mathbf { j } - 2 \mathbf { k } )$$

1. Find the values of \(a\) and \(b\). The point \(P\) lies on \(l _ { 1 }\) and is such that \(O P\) is perpendicular to \(l _ { 1 }\), where \(O\) is the origin.
2. Find the position vector of point \(P\). Given that \(B\) has coordinates \(( 5,15,1 )\),
3. show that the points \(A , P\) and \(B\) are collinear and find the ratio \(A P : P B\).

1. With respect to a fixed origin \(O\), the line \(l\) has equation

$$\mathbf { r } = \left( \begin{array} { c } 13 \\ 8 \\ 1 \end{array} \right) + \lambda \left( \begin{array} { r } 2 \\ 2 \\ - 1 \end{array} \right) \text {, where } \lambda \text { is a scalar parameter. }$$ The point \(A\) lies on \(l\) and has coordinates ( \(3 , - 2,6\) ).
The point \(P\) has position vector ( \(- p \mathbf { i } + 2 p \mathbf { k }\) ) relative to \(O\), where \(p\) is a constant.
Given that vector \(\overrightarrow { P A }\) is perpendicular to \(l\),

1. find the value of \(p\). Given also that \(B\) is a point on \(l\) such that \(\angle B P A = 45 ^ { \circ }\),
2. find the coordinates of the two possible positions of \(B\). \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) Question 8 continued

6. With respect to a fixed origin, the point \(A\) with position vector \(\mathbf { i } + 2 \mathbf { j } + 3 \mathbf { k }\) lies on the line \(l _ { 1 }\) with equation $$\mathbf { r } = \left( \begin{array} { l } 1 \\ 2 \\ 3 \end{array} \right) + \lambda \left( \begin{array} { r } 0 \\ 2 \\ - 1 \end{array} \right) , \quad \text { where } \lambda \text { is a scalar parameter, }$$ and the point \(B\) with position vector \(4 \mathbf { i } + p \mathbf { j } + 3 \mathbf { k }\), where \(p\) is a constant, lies on the line \(l _ { 2 }\) with equation $$\mathbf { r } = \left( \begin{array} { l } 7 \\ 0 \\ 7 \end{array} \right) + \mu \left( \begin{array} { r } 3 \\ - 5 \\ 4 \end{array} \right) , \quad \text { where } \mu \text { is a scalar parameter. }$$

1. Find the value of the constant \(p\).
2. Show that \(l _ { 1 }\) and \(l _ { 2 }\) intersect and find the position vector of their point of intersection, \(C\).
3. Find the size of the angle \(A C B\), giving your answer in degrees to 3 significant figures.
4. Find the area of the triangle \(A B C\), giving your answer to 3 significant figures. \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \section*{Question 6 continued} \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\)

8. With respect to a fixed origin \(O\), the line \(l _ { 1 }\) is given by the equation $$\mathbf { r } = \left( \begin{array} { r } 8 \\ 1 \\ - 3 \end{array} \right) + \mu \left( \begin{array} { r } - 5 \\ 4 \\ 3 \end{array} \right)$$ where \(\mu\) is a scalar parameter.
The point \(A\) lies on \(l _ { 1 }\) where \(\mu = 1\)

1. Find the coordinates of \(A\). The point \(P\) has position vector \(\left( \begin{array} { l } 1 \\ 5 \\ 2 \end{array} \right)\).
   The line \(l _ { 2 }\) passes through the point \(P\) and is parallel to the line \(l _ { 1 }\)
2. Write down a vector equation for the line \(l _ { 2 }\)
3. Find the exact value of the distance \(A P\). Give your answer in the form \(k \sqrt { 2 }\), where \(k\) is a constant to be determined. The acute angle between \(A P\) and \(l _ { 2 }\) is \(\theta\).
4. Find the value of \(\cos \theta\) A point \(E\) lies on the line \(l _ { 2 }\) Given that \(A P = P E\),
5. find the area of triangle \(A P E\),
6. find the coordinates of the two possible positions of \(E\).

7. The point \(A\) with coordinates ( \(- 3,7,2\) ) lies on a line \(l _ { 1 }\) The point \(B\) also lies on the line \(l _ { 1 }\) Given that \(\quad \overrightarrow { A B } = \left( \begin{array} { r } 4 \\ - 6 \\ 2 \end{array} \right)\),

1. find the coordinates of point \(B\). The point \(P\) has coordinates ( \(9,1,8\) )
2. Find the cosine of the angle \(P A B\), giving your answer as a simplified surd.
3. Find the exact area of triangle \(P A B\), giving your answer in its simplest form. The line \(l _ { 2 }\) passes through the point \(P\) and is parallel to the line \(l _ { 1 }\)
4. Find a vector equation for the line \(l _ { 2 }\) The point \(Q\) lies on the line \(l _ { 2 }\) Given that the line segment \(A P\) is perpendicular to the line segment \(B Q\),
5. find the coordinates of the point \(Q\).

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

• the point \(A\) has position vector \(4 \mathbf { i } + 8 \mathbf { j } + \mathbf { k }\)
• the point \(B\) has position vector \(5 \mathbf { i } + 6 \mathbf { j } + 3 \mathbf { k }\)
• the point \(P\) has position vector \(2 \mathbf { i } - 2 \mathbf { j } + \mathbf { k }\)

The straight line \(l\) passes through \(A\) and \(B\).

1. Find a vector equation for \(l\). The point \(C\) lies on \(l\) so that \(P C\) is perpendicular to \(l\).
2. Find the coordinates of \(C\). The point \(P ^ { \prime }\) is the reflection of \(P\) in the line \(l\).
3. Find the coordinates of \(P ^ { \prime }\)
4. Hence find \(\left| \overrightarrow { P P ^ { \prime } } \right|\), giving your answer as a simplified surd.

7. [In this question, the unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are due east and due north respectively. Position vectors are relative to a fixed origin \(O\).] A boat \(P\) is moving with constant velocity \(( - 4 \mathbf { i } + 8 \mathbf { j } ) \mathrm { km } \mathrm { h } ^ { - 1 }\).

1. Calculate the speed of \(P\). When \(t = 0\), the boat \(P\) has position vector \(( 2 \mathbf { i } - 8 \mathbf { j } ) \mathrm { km }\). At time \(t\) hours, the position vector of \(P\) is \(\mathbf { p ~ k m }\).
2. Write down \(\mathbf { p }\) in terms of \(t\). A second boat \(Q\) is also moving with constant velocity. At time \(t\) hours, the position vector of \(Q\) is \(\mathbf { q } \mathrm { km }\), where $$\mathbf { q } = 18 \mathbf { i } + 12 \mathbf { j } - t ( 6 \mathbf { i } + 8 \mathbf { j } )$$ Find
3. the value of \(t\) when \(P\) is due west of \(Q\),
4. the distance between \(P\) and \(Q\) when \(P\) is due west of \(Q\).

7. [In this question, the horizontal unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are directed due east and north respectively] A mountain rescue post \(O\) receives a distress call via a mobile phone from a walker who has broken a leg and cannot move. The walker says he is by a pipeline and he can also see a radio mast which he believes to be south-west of him. The pipeline is known to run north-south for a long distance through the point with position vector \(6 \mathbf { i } \mathrm {~km}\), relative to \(O\). The radio mast is known to be at the point with position vector \(2 \mathbf { j } \mathrm {~km}\), relative to \(O\).

1. Using the information supplied by the walker, write down his position vector in the form \(( a \mathbf { i } + b \mathbf { j } )\). The rescue party moves at a horizontal speed of \(5 \mathrm {~km} \mathrm {~h} ^ { - 1 }\). The leader of the party wants to give the walker and idea of how long it will take to for the rescue party to arrive.
2. Calculate how long it will take for the rescue party to reach the walker's estimated position. The rescue party sets out and walks straight towards the walker's estimated position at a constant horizontal speed of \(5 \mathrm {~km} \mathrm {~h} ^ { - 1 }\). After the party has travelled for one hour, the walker rings again. He is very apologetic and says that he now realises that the radio mask is in fact north-west of his position
3. Find the position vector of the walker.
4. Find in degrees to one decimal place, the bearing on which the rescue party should now travel in order to reach the walker directly. \section*{END}

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 with constant velocity \(( - 12 \mathbf { i } + 7.5 \mathbf { j } ) \mathrm { km } \mathrm { h } ^ { - 1 }\).

1. Find the direction in which \(S\) is moving, giving your answer as a bearing. At time \(t\) hours after noon, the position vector of \(S\) is \(\mathbf { s } \mathrm { km }\). When \(t = 0 , \mathbf { s } = 40 \mathbf { i } - 6 \mathbf { j }\).
2. Write down \(\mathbf { s }\) in terms of \(t\). A fixed beacon \(B\) is at the point with position vector \(( 7 \mathbf { i } + 12.5 \mathbf { j } ) \mathrm { km }\).
3. Find the distance of \(S\) from \(B\) when \(t = 3\)
4. Find the distance of \(S\) from \(B\) when \(S\) is due north of \(B\).

1. \hspace{0pt} [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 \(O\).]

At midnight, a ship \(S\) is at the point with position vector ( \(19 \mathbf { i } + 22 \mathbf { j }\) )km
The ship travels with constant velocity \(( 12 \mathbf { i } - 16 \mathbf { j } ) \mathrm { km } \mathrm { h } ^ { - 1 }\)

1. Find the speed of \(S\). At time \(t\) hours after midnight, the position vector of \(S\) is \(\mathbf { s } \mathrm { km }\).
2. Find an expression for \(\mathbf { s }\) in terms of \(\mathbf { i } , \mathbf { j }\) and \(t\). A lighthouse stands on a small rocky island. The lighthouse is modelled as being at the point with position vector \(( 26 \mathbf { i } + 15 \mathbf { j } ) \mathrm { km }\). It is not safe for ships to be within 1.3 km of the lighthouse.
3. 1. Find the value of \(t\) when \(S\) is closest to the lighthouse.
   2. Hence determine whether it is safe for \(S\) to continue its course.

5. [In this question \(\mathbf { i }\) and \(\mathbf { j }\) are perpendicular horizontal unit vectors and position vectors are given relative to a fixed origin \(O\).] A particle \(P\) is moving in a straight line with constant velocity. At 9 am, the position vector of \(P\) is \(( 7 \mathbf { i } + 5 \mathbf { j } ) \mathrm { km }\) and at 9.20 am , the position vector of \(P\) is \(6 \mathbf { i } \mathrm {~km}\). At time \(t\) hours after 9 am , the position vector of \(P\) is \(\mathbf { r } _ { P } \mathrm {~km}\).

1. Find, in \(\mathrm { kmh } ^ { - 1 }\), the speed of \(P\).
2. Show that \(\mathbf { r } _ { P } = ( 7 - 3 t ) \mathbf { i } + ( 5 - 15 t ) \mathbf { j }\).
3. Find the value of \(t\) when \(\mathbf { r } _ { P }\) is parallel to \(16 \mathbf { i } + 5 \mathbf { j }\). The position vector of another particle \(Q\), at time \(t\) hours after 9 am , is \(\mathbf { r } _ { Q } \mathrm {~km}\), where \(\mathbf { r } _ { Q } = ( 5 + 2 t ) \mathbf { i } + ( - 3 + 5 t ) \mathbf { j }\)
4. Show that \(P\) and \(Q\) will collide and find the position vector of the point of collision.

8. [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 boats, \(P\) and \(Q\), are moving with constant velocities.
The velocity of \(P\) is \(15 \mathbf { i } \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and the velocity of \(Q\) is \(( 20 \mathbf { i } - 20 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\)

1. Find the direction in which \(Q\) is travelling, giving your answer as a bearing. The boats are modelled as particles.
   At time \(t = 0 , P\) is at the origin \(O\) and \(Q\) is at the point with position vector \(200 \mathbf { j } \mathrm {~m}\). At time \(t\) seconds, the position vector of \(P\) is \(\mathbf { p m }\) and the position vector of \(Q\) is \(\mathbf { q m }\).
2. Show that $$\overrightarrow { P Q } = [ 5 t \mathbf { i } + ( 200 - 20 t ) \mathbf { j } ] \mathrm { m }$$
3. Find the bearing of \(P\) from \(Q\) when \(t = 10\)
4. Find the distance between \(P\) and \(Q\) when \(Q\) is north east of \(P\)
5. Find the times when \(P\) and \(Q\) are 200 m apart.

8. \begin{figure}[h] \end{figure} A square floor space \(A B C D\), with centre \(O\), is modelled as a flat horizontal surface measuring 50 m by 50 m , as shown in Figure 5 .
The horizontal unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are in the direction of \(\overrightarrow { A B }\) and \(\overrightarrow { A D }\) respectively.
All position vectors are given relative to \(O\).
A small robot \(R\) is programmed to travel across the floor at a constant velocity.

• At time \(t = 0 , R\) is at the point with position vector ( \(- 2 \mathbf { i } + \mathbf { j }\) ) m
• At time \(t = 11 \mathrm {~s} , R\) is at the point with position vector \(( 9 \mathbf { i } + 23 \mathbf { j } ) \mathrm { m }\)
• At time \(t\) seconds, the position vector of \(R\) is \(\mathbf { r }\) metres
  1. Find, in terms of \(t\), i and j, an expression for \(\mathbf { r }\)

A second robot \(S\) is at the point \(C\).

$$\overrightarrow { S R } = [ ( 2 t - 27 ) \mathbf { i } + ( 3 t - 24 ) \mathbf { j } ] \mathbf { m }$$

Find the time when the distance between \(R\) and \(S\) is a minimum.

1. The position vector, \(\mathbf { r }\) metres, of a particle \(P\) at time \(t\) seconds, relative to a fixed origin \(O\), is given by

$$\mathbf { r } = ( t - 3 ) \mathbf { i } + ( 1 - 2 t ) \mathbf { j }$$

1. Find, to the nearest degree, the size of the angle between \(\mathbf { r }\) and the vector \(\mathbf { j }\), when \(t = 2\)
2. Find the values of \(t\) for which the distance of \(P\) from \(O\) is 2.5 m .

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 \(\mathbf { i }\).
   (2 marks)
   Three seconds after it passes \(B\) the particle \(P\) reaches the point \(C\).
3. Find, in m to one decimal place, the distance \(O C\).

9 The points \(A , B\) and \(C\) have coordinates \(( 5,1 ) , ( p , 7 )\) and \(( 8,2 )\) respectively.

1. Given that the distance between points \(A\) and \(B\) is twice the distance between points \(A\) and \(C\), calculate the possible values of \(p\).
2. Given also that the line passing through \(A\) and \(B\) has equation \(y = 3 x - 14\), find the coordinates of the mid-point of \(A B\).

9 The points \(A\) and \(B\) have coordinates \(( - 5 , - 2 )\) and \(( 3,1 )\) respectively.

1. Find the equation of the line \(A B\), giving your answer in the form \(a x + b y + c = 0\).
2. Find the coordinates of the mid-point of \(A B\). The point \(C\) has coordinates (-3,4).
3. Calculate the length of \(A C\), giving your answer in simplified surd form.
4. Determine whether the line \(A C\) is perpendicular to the line \(B C\), showing all your working.

9

1. Find the gradient of the line \(l _ { 1 }\) which has equation \(4 x - 3 y + 5 = 0\).
2. Find an equation of the line \(l _ { 2 }\), which passes through the point ( 1,2 ) and which is perpendicular to the line \(l _ { 1 }\), giving your answer in the form \(a x + b y + c = 0\). The line \(l _ { 1 }\) crosses the \(x\)-axis at \(P\) and the line \(l _ { 2 }\) crosses the \(y\)-axis at \(Q\).
3. Find the coordinates of the mid-point of \(P Q\).
4. Calculate the length of \(P Q\), giving your answer in the form \(\frac { \sqrt { } a } { b }\), where \(a\) and \(b\) are integers.

9 The circle with equation \(x ^ { 2 } + y ^ { 2 } - 6 x - k = 0\) has radius 4 .

1. Find the centre of the circle and the value of k . The points \(\mathrm { A } ( 3 , \mathrm { a } )\) and \(\mathrm { B } ( - 1,0 )\) lie on the circumference of the circle, with \(\mathrm { a } > 0\).
2. Calculate the length of \(A B\), giving your answer in simplified surd form.
3. Find an equation for the line \(A B\).

1. The straight line \(l\) has the equation \(x - 2 y = 12\) and meets the coordinate axes at the points \(A\) and \(B\).

Find the distance of the mid-point of \(A B\) from the origin, giving your answer in the form \(k \sqrt { 5 }\).

1 Point A has coordinates ( 4,7 ) and point B has coordinates ( 2,1 ).

1. Find the equation of the line through A and B .
2. Point C has coordinates \(( - 1,2 )\). Show that angle \(\mathrm { ABC } = 90 ^ { \circ }\) and calculate the area of triangle ABC .
3. Find the coordinates of \(D\), the midpoint of AC. Explain also how you can tell, without having to work it out, that \(\mathrm { A } , \mathrm { B }\) and C are all the same distance from D.

4. A particle \(P\) of mass 0.4 kg is moving under the action of a single force \(\mathbf { F }\) newtons. At time \(t\) seconds, the velocity of \(P , \mathbf { v } \mathrm {~m} \mathrm {~s} ^ { - 1 }\), is given by $$\mathbf { v } = ( 6 t + 4 ) \mathbf { i } + \left( t ^ { 2 } + 3 t \right) \mathbf { j } .$$ When \(t = 0 , P\) is at the point with position vector \(( - 3 \mathbf { i } + 4 \mathbf { j } ) \mathrm { m }\). When \(t = 4 , P\) is at the point \(S\).

1. Calculate the magnitude of \(\mathbf { F }\) when \(t = 4\).
2. Calculate the distance \(O S\).