Perpendicular from point to line

1. (i) Relative to a fixed origin \(O\), the line \(l _ { 1 }\) is given by the equation

$$l _ { 1 } : \mathbf { r } = \left( \begin{array} { r } - 5 \\ 1 \\ 6 \end{array} \right) + \lambda \left( \begin{array} { r } 2 \\ - 3 \\ 1 \end{array} \right) \text { where } \lambda \text { is a scalar parameter. }$$ The point \(P\) lies on \(l _ { 1 }\). Given that \(\overrightarrow { O P }\) is perpendicular to \(l _ { 1 }\), calculate the coordinates of \(P\).
(ii) Relative to a fixed origin \(O\), the line \(l _ { 2 }\) is given by the equation $$l _ { 2 } : \mathbf { r } = \left( \begin{array} { r } 4 \\ - 3 \\ 12 \end{array} \right) + \mu \left( \begin{array} { r } 5 \\ - 3 \\ 4 \end{array} \right) \text { where } \mu \text { is a scalar parameter. }$$ The point \(A\) does not lie on \(l _ { 2 }\). Given that the vector \(\overrightarrow { O A }\) is parallel to the line \(l _ { 2 }\) and \(| \overrightarrow { O A } | = \sqrt { 2 }\) units, calculate the possible position vectors of the point \(A\).

8. With respect to a fixed origin \(O\), the lines \(l _ { 1 }\) and \(l _ { 2 }\) are given by the equations $$l _ { 1 } : \mathbf { r } = \left( \begin{array} { r } 1 \\ - 3 \\ 2 \end{array} \right) + \lambda \left( \begin{array} { l } 1 \\ 2 \\ 3 \end{array} \right) , \quad l _ { 2 } : \mathbf { r } = \left( \begin{array} { l } 6 \\ 4 \\ 1 \end{array} \right) + \mu \left( \begin{array} { r } 1 \\ 1 \\ - 1 \end{array} \right)$$ where \(\lambda\) and \(\mu\) are scalar parameters.

1. Show that \(l _ { 1 }\) and \(l _ { 2 }\) do not meet. The point \(P\) is on \(l _ { 1 }\) where \(\lambda = 0\), and the point \(Q\) is on \(l _ { 2 }\) where \(\mu = - 1\)
2. Find the acute angle between the line segment \(P Q\) and \(l _ { 1 }\), giving your answer in degrees to 2 decimal places.
3. Find the shortest distance from the point \(Q\) to the line \(l _ { 1 }\), giving your answer to 3 significant figures.

6. The line \(l _ { 1 }\) has vector equation $$\mathbf { r } = 8 \mathbf { i } + 12 \mathbf { j } + 14 \mathbf { k } + \lambda ( \mathbf { i } + \mathbf { j } - \mathbf { k } ) ,$$ where \(\lambda\) is a parameter. The point \(A\) has coordinates (4, 8, a), where \(a\) is a constant. The point \(B\) has coordinates ( \(b , 13,13\) ), where \(b\) is a constant. Points \(A\) and \(B\) lie on the line \(l _ { 1 }\).

1. Find the values of \(a\) and \(b\). Given that the point \(O\) is the origin, and that the point \(P\) lies on \(l _ { 1 }\) such that \(O P\) is perpendicular to \(l _ { 1 }\),
2. find the coordinates of \(P\).
3. Hence find the distance \(O P\), giving your answer as a simplified surd.

7. With respect to a fixed origin \(O\), the lines \(l _ { 1 }\) and \(l _ { 2 }\) are given by the equations $$\begin{aligned} & l _ { 1 } : \mathbf { r } = ( 9 \mathbf { i } + 13 \mathbf { j } - 3 \mathbf { k } ) + \lambda ( \mathbf { i } + 4 \mathbf { j } - 2 \mathbf { k } ) \\ & l _ { 2 } : \mathbf { r } = ( 2 \mathbf { i } - \mathbf { j } + \mathbf { k } ) + \mu ( 2 \mathbf { i } + \mathbf { j } + \mathbf { k } ) \end{aligned}$$ where \(\lambda\) and \(\mu\) are scalar parameters.

1. Given that \(l _ { 1 }\) and \(l _ { 2 }\) meet, find the position vector of their point of intersection.
2. Find the acute angle between \(l _ { 1 }\) and \(l _ { 2 }\), giving your answer in degrees to 1 decimal place. Given that the point \(A\) has position vector \(4 \mathbf { i } + 16 \mathbf { j } - 3 \mathbf { k }\) and that the point \(P\) lies on \(l _ { 1 }\) such that \(A P\) is perpendicular to \(l _ { 1 }\),
3. find the exact coordinates of \(P\).

7. Relative to a fixed origin \(O\), the point \(A\) has position vector \(( 8 \mathbf { i } + 13 \mathbf { j } - 2 \mathbf { k } )\), the point \(B\) has position vector ( \(10 \mathbf { i } + 14 \mathbf { j } - 4 \mathbf { k }\) ), and the point \(C\) has position vector \(( 9 \mathbf { i } + 9 \mathbf { j } + 6 \mathbf { k } )\). The line \(l\) passes through the points \(A\) and \(B\).

1. Find a vector equation for the line \(l\).
2. Find \(| \overrightarrow { C B } |\).
3. Find the size of the acute angle between the line segment \(C B\) and the line \(l\), giving your answer in degrees to 1 decimal place.
4. Find the shortest distance from the point \(C\) to the line \(l\). The point \(X\) lies on \(l\). Given that the vector \(\overrightarrow { C X }\) is perpendicular to \(l\),
5. find the area of the triangle \(C X B\), giving your answer to 3 significant figures.

6. With respect to a fixed origin \(O\), the lines \(l _ { 1 }\) and \(l _ { 2 }\) are given by the equations $$l _ { 1 } : \quad \mathbf { r } = \left( \begin{array} { r } 6
- 3
- 2 \end{array} \right) + \lambda \left( \begin{array} { r } - 1
2
3 \end{array} \right) , \quad l _ { 2 } : \mathbf { r } = \left( \begin{array} { r } - 5

1. Relative to a fixed origin \(O\), the point \(A\) has position vector \(( 10 \mathbf { i } + 2 \mathbf { j } + 3 \mathbf { k } )\), and the point \(B\) has position vector \(( 8 \mathbf { i } + 3 \mathbf { j } + 4 \mathbf { k } )\).

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

1. Find the vector \(\overrightarrow { A B }\).
2. Find a vector equation for the line \(l\). The point \(C\) has position vector \(( 3 \mathbf { i } + 12 \mathbf { j } + 3 \mathbf { k } )\).
   The point \(P\) lies on \(l\). Given that the vector \(\overrightarrow { C P }\) is perpendicular to \(l\),
3. find the position vector of the point \(P\).

5 \end{array} \right) + \mu \left( \begin{array} { r } 5
- 4
- 3 \end{array} \right)$$ where \(\mu\) is a scalar parameter.\\ The point \(A\) is on \(l _ { 1 }\) where \(\mu = 2\).

1. Write down the coordinates of \(A\). The acute angle between \(O A\) and \(l _ { 1 }\) is \(\theta\), where \(O\) is the origin.
2. Find the value of \(\cos \theta\). The point \(B\) is such that \(\overrightarrow { O B } = 3 \overrightarrow { O A }\).\\ The line \(l _ { 2 }\) passes through the point \(B\) and is parallel to the line \(l _ { 1 }\).
3. Find a vector equation of \(l _ { 2 }\).
4. Find the length of \(O B\), giving your answer as a simplified surd. The point \(X\) lies on \(l _ { 2 }\). Given that the vector \(\overrightarrow { O X }\) is perpendicular to \(l _ { 2 }\),
5. find the length of \(O X\), giving your answer to 3 significant figures.\\ 5. The curve \(C\) has the equation $$\sin ( \pi y ) - y - x ^ { 2 } y = - 5 , \quad x > 0$$
6. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(x\) and \(y\). The point \(P\) with coordinates \(( 2,1 )\) lies on \(C\).
   The tangent to \(C\) at \(P\) meets the \(x\)-axis at the point \(A\).
7. Find the exact value of the \(x\)-coordinate of \(A\).

8
5 \end{array} \right) + \mu \left( \begin{array} { r } 5
- 4
- 3 \end{array} \right)$$ where \(\mu\) is a scalar parameter.\\ The point \(A\) is on \(l _ { 1 }\) where \(\mu = 2\).

1. Write down the coordinates of \(A\). The acute angle between \(O A\) and \(l _ { 1 }\) is \(\theta\), where \(O\) is the origin.
2. Find the value of \(\cos \theta\). The point \(B\) is such that \(\overrightarrow { O B } = 3 \overrightarrow { O A }\).\\ The line \(l _ { 2 }\) passes through the point \(B\) and is parallel to the line \(l _ { 1 }\).
3. Find a vector equation of \(l _ { 2 }\).
4. Find the length of \(O B\), giving your answer as a simplified surd. The point \(X\) lies on \(l _ { 2 }\). Given that the vector \(\overrightarrow { O X }\) is perpendicular to \(l _ { 2 }\),
5. find the length of \(O X\), giving your answer to 3 significant figures.\\ 5. The curve \(C\) has the equation $$\sin ( \pi y ) - y - x ^ { 2 } y = - 5 , \quad x > 0$$
6. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(x\) and \(y\). The point \(P\) with coordinates \(( 2,1 )\) lies on \(C\).\\ The tangent to \(C\) at \(P\) meets the \(x\)-axis at the point \(A\).
7. Find the exact value of the \(x\)-coordinate of \(A\).\\ 6. (i) (a) Express \(\frac { 7 x } { ( x + 3 ) ( 2 x - 1 ) }\) in partial fractions.
8. Given that \(x > \frac { 1 } { 2 }\), find $$\int \frac { 7 x } { ( x + 3 ) ( 2 x - 1 ) } d x$$ (ii) Using the substitution \(u ^ { 3 } = x\), or otherwise, find $$\int \frac { 1 } { x + x ^ { \frac { 1 } { 3 } } } d x , \quad x > 0$$ 7. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{b0d21f0f-f5f6-4ca5-8e3e-98aee0d9db7a-11_703_1164_373_492} \captionsetup{labelformat=empty} \caption{Figure 2}
   \end{figure} Figure 2 shows a sketch of part of the curve \(C\) with parametric equations $$x = \tan \theta , \quad y = 1 + 2 \cos 2 \theta , \quad 0 \leqslant \theta < \frac { \pi } { 2 }$$ The curve \(C\) crosses the \(x\)-axis at \(( \sqrt { } 3,0 )\). The finite shaded region \(S\) shown in Figure 2 is bounded by \(C\), the line \(x = 1\) and the \(x\)-axis. This shaded region is rotated through \(2 \pi\) radians about the \(x\)-axis to form a solid of revolution.
9. Show that the volume of the solid of revolution formed is given by the integral $$k \int _ { \frac { \pi } { 4 } } ^ { \frac { \pi } { 3 } } \left( 16 \cos ^ { 2 } \theta - 8 + \sec ^ { 2 } \theta \right) d \theta$$ where \(k\) is a constant.
10. Hence, use integration to find the exact value for this volume.\\ 8. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{b0d21f0f-f5f6-4ca5-8e3e-98aee0d9db7a-13_869_545_312_811} \captionsetup{labelformat=empty} \caption{Figure 3}
    \end{figure} Figure 3 shows a large vertical cylindrical tank containing a liquid. The radius of the circular cross-section of the tank is 40 cm . At time \(t\) minutes, the depth of liquid in the tank is \(h\) centimetres. The liquid leaks from a hole \(P\) at the bottom of the tank. The liquid leaks from the tank at a rate of \(32 \pi \sqrt { } h \mathrm {~cm} ^ { 3 } \mathrm {~min} ^ { - 1 }\).
11. Show that at time \(t\) minutes, the height \(h \mathrm {~cm}\) of liquid in the tank satisfies the differential equation $$\frac { \mathrm { d } h } { \mathrm {~d} t } = - 0.02 \sqrt { } h$$
12. Find the time taken, to the nearest minute, for the depth of liquid in the tank to decrease from 100 cm to 50 cm .
    
    \includegraphics[max width=\textwidth, alt={}]{b0d21f0f-f5f6-4ca5-8e3e-98aee0d9db7a-14_2639_1834_214_217}

8
5
- 2 \end{array} \right) + \mu \left( \begin{array} { r } 3
4
- 5 \end{array} \right)$$ where \(\lambda\) and \(\mu\) are scalar parameters and \(p\) is a constant. The lines \(l _ { 1 }\) and \(l _ { 2 }\) intersect at the point \(A\).

1. Find the coordinates of \(A\).
2. Find the value of the constant \(p\).
3. Find the acute angle between \(l _ { 1 }\) and \(l _ { 2 }\), giving your answer in degrees to 2 decimal places. The point \(B\) lies on \(l _ { 2 }\) where \(\mu = 1\)
4. Find the shortest distance from the point \(B\) to the line \(l _ { 1 }\), giving your answer to 3 significant figures.
   1. A curve \(C\) has parametric equations
   $$x = 4 t + 3 , \quad y = 4 t + 8 + \frac { 5 } { 2 t } , \quad t \neq 0$$
5. Find the value of \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) at the point on \(C\) where \(t = 2\), giving your answer as a fraction in its simplest form.
6. Show that the cartesian equation of the curve \(C\) can be written in the form $$y = \frac { x ^ { 2 } + a x + b } { x - 3 } , \quad x \neq 3$$ where \(a\) and \(b\) are integers to be determined.\\ 6. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{89d4a7a5-3f4f-4d16-b14e-a27243cedd78-11_666_993_244_392} \captionsetup{labelformat=empty} \caption{Diagram not to scale}
   \end{figure} Figure 2 Figure 2 shows a sketch of the curve with equation \(y = \sqrt { ( 3 - x ) ( x + 1 ) } , 0 \leqslant x \leqslant 3\)\\ The finite region \(R\), shown shaded in Figure 2, is bounded by the curve, the \(x\)-axis, and the \(y\)-axis.
7. Use the substitution \(x = 1 + 2 \sin \theta\) to show that $$\int _ { 0 } ^ { 3 } \sqrt { ( 3 - x ) ( x + 1 ) } d x = k \int _ { - \frac { \pi } { 6 } } ^ { \frac { \pi } { 2 } } \cos ^ { 2 } \theta d \theta$$ where \(k\) is a constant to be determined.
8. Hence find, by integration, the exact area of \(R\). 7. (a) Express \(\frac { 2 } { P ( P - 2 ) }\) in partial fractions. A team of biologists is studying a population of a particular species of animal. The population is modelled by the differential equation $$\frac { \mathrm { d } P } { \mathrm {~d} t } = \frac { 1 } { 2 } P ( P - 2 ) \cos 2 t , t \geqslant 0$$ where \(P\) is the population in thousands, and \(t\) is the time measured in years since the start of the study. Given that \(P = 3\) when \(t = 0\),
9. solve this differential equation to show that $$P = \frac { 6 } { 3 - \mathrm { e } ^ { \frac { 1 } { 2 } \sin 2 t } }$$
10. find the time taken for the population to reach 4000 for the first time. Give your answer in years to 3 significant figures.\\ 8. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{89d4a7a5-3f4f-4d16-b14e-a27243cedd78-15_696_1418_287_262} \captionsetup{labelformat=empty} \caption{Figure 3}
    \end{figure} Figure 3 shows a sketch of part of the curve \(C\) with equation $$y = 3 ^ { x }$$ The point \(P\) lies on \(C\) and has coordinates \(( 2,9 )\).
    The line \(l\) is a tangent to \(C\) at \(P\). The line \(l\) cuts the \(x\)-axis at the point \(Q\).
11. Find the exact value of the \(x\) coordinate of \(Q\). The finite region \(R\), shown shaded in Figure 3, is bounded by the curve \(C\), the \(x\)-axis, the \(y\)-axis and the line \(l\). This region \(R\) is rotated through \(360 ^ { \circ }\) about the \(x\)-axis.
12. Use integration to find the exact value of the volume of the solid generated. Give your answer in the form \(\frac { p } { q }\) where \(p\) and \(q\) are exact constants.
    [0pt] [You may assume the formula \(V = \frac { 1 } { 3 } \pi r ^ { 2 } h\) for the volume of a cone.]

10 The position vectors of the points \(P\) and \(Q\) with respect to an origin \(O\) are \(5 \mathbf { i } + 2 \mathbf { j } - 9 \mathbf { k }\) and \(4 \mathbf { i } + 4 \mathbf { j } - 6 \mathbf { k }\) respectively.

1. Find a vector equation for the line \(P Q\). The position vector of the point \(T\) is \(\mathbf { i } + 2 \mathbf { j } - \mathbf { k }\).
2. Write down a vector equation for the line \(O T\) and show that \(O T\) is perpendicular to \(P Q\). It is given that \(O T\) intersects \(P Q\).
3. Find the position vector of the point of intersection of \(O T\) and \(P Q\).
4. Hence find the perpendicular distance from \(O\) to \(P Q\), giving your answer in an exact form.

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

1. Find a vector equation of the line passing through \(A\) and \(B\).
2. Find the position vector of the point \(P\) on \(A B\) such that \(O P\) is perpendicular to \(A B\).

6. Relative to a fixed origin, \(O\), the line \(l\) has the equation $$\mathbf { r } = \left( \begin{array} { c } 1 \\ p \\ - 5 \end{array} \right) + \lambda \left( \begin{array} { c } 3 \\ - 1 \\ q \end{array} \right)$$ where \(p\) and \(q\) are constants and \(\lambda\) is a scalar parameter.
Given that the point \(A\) with coordinates \(( - 5,9 , - 9 )\) lies on \(l\),

1. find the values of \(p\) and \(q\),
2. show that the point \(B\) with coordinates \(( 25 , - 1,11 )\) also lies on \(l\). The point \(C\) lies on \(l\) and is such that \(O C\) is perpendicular to \(l\).
3. Find the coordinates of \(C\).
4. Find the ratio \(A C : C B\)

9 The equation of a straight line \(l\) is \(\mathbf { r } = \left( \begin{array} { l } 3 \\ 1 \\ 1 \end{array} \right) + t \left( \begin{array} { r } 1 \\ - 1 \\ 2 \end{array} \right) . O\) is the origin.

1. The point \(P\) on \(l\) is given by \(t = 1\). Calculate the acute angle between \(O P\) and \(l\).
2. Find the position vector of the point \(Q\) on \(l\) such that \(O Q\) is perpendicular to \(l\).
3. Find the length of \(O Q\).

6
4 \end{array} \right) + s \left( \begin{array} { l } 3
2
1 \end{array} \right) \quad \text { and } \quad \mathbf { r } = \left( \begin{array} { l } 1
0
0 \end{array} \right) + t \left( \begin{array} { r } 0
1
- 1 \end{array} \right)$$ respectively.

1. Show that \(l _ { 1 }\) and \(l _ { 2 }\) are skew.
2. Find the acute angle between \(l _ { 1 }\) and \(l _ { 2 }\).
3. The point \(A\) lies on \(l _ { 1 }\) and \(O A\) is perpendicular to \(l _ { 1 }\), where \(O\) is the origin. Find the position vector of \(A\). 6 Find the coefficient of \(x ^ { 2 }\) in the expansion in ascending powers of \(x\) of $$\sqrt { \frac { 1 + a x } { 4 - x } } ,$$ giving your answer in terms of \(a\).

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.}

The line \(l _ { 1 }\) passes through the point \(A\) whose position vector is \(4 \mathbf { i } + 7 \mathbf { j } - \mathbf { k }\) and is parallel to the vector \(3 \mathbf { i } + 2 \mathbf { j } - \mathbf { k }\). The line \(l _ { 2 }\) passes through the point \(B\) whose position vector is \(\mathbf { i } + 7 \mathbf { j } + 11 \mathbf { k }\) and is parallel to the vector \(\mathbf { i } - 6 \mathbf { j } - 2 \mathbf { k }\). The points \(P\) on \(l _ { 1 }\) and \(Q\) on \(l _ { 2 }\) are such that \(P Q\) is perpendicular to both \(l _ { 1 }\) and \(l _ { 2 }\). Find the position vectors of \(P\) and \(Q\). Find the shortest distance between the line through \(A\) and \(B\) and the line through \(P\) and \(Q\), giving your answer correct to 3 significant figures.

5 The line through points \(A ( 8 , - 7 , - 2 )\) and \(B ( 11 , - 9,0 )\) is denoted by \(L _ { 1 }\).

1. Find a vector equation for \(L _ { 1 }\).
2. Determine whether the point \(( 26 , - 19 , - 14 )\) lies on \(L _ { 1 }\). The line \(L _ { 2 }\) passes through the origin, \(O\), and intersects \(L _ { 1 }\) at the point \(C\). The lines \(L _ { 1 }\) and \(L _ { 2 }\) are perpendicular.
3. By using the fact that \(C\) lies on \(L _ { 1 }\), find a vector equation for \(L _ { 2 }\).
4. Hence find the shortest distance from \(O\) to \(L _ { 1 }\).

4. Relative to a fixed origin \(O\), the point \(A\) has position vector \(4 \mathbf { i } + 8 \mathbf { j } - \mathbf { k }\), and the point \(B\) has position vector \(7 \mathbf { i } + 14 \mathbf { j } + 5 \mathbf { k }\).

1. Find the vector \(\overrightarrow { A B }\).
2. Calculate the cosine of \(\angle O A B\).
3. Show that, for all values of \(\lambda\), the point P with position vector \(\lambda \mathbf { i } + 2 \lambda \mathbf { j } + ( 2 \lambda - 9 ) \mathbf { k }\) lies on the line through \(A\) and \(B\).
4. Find the value of \(\lambda\) for which \(O P\) is perpendicular to \(A B\).
5. Hence find the coordinates of the foot of the perpendicular from \(O\) to \(A B\).

8. Referred to an origin \(O\), the points \(A , B\) and \(C\) have position vectors ( \(9 \mathbf { i } - 2 \mathbf { j } + \mathbf { k }\) ), \(( 6 \mathbf { i } + 2 \mathbf { j } + 6 \mathbf { k } )\) and \(( 3 \mathbf { i } + p \mathbf { j } + q \mathbf { k } )\) respectively, where \(p\) and \(q\) are constants.

1. Find, in vector form, an equation of the line \(l\) which passes through \(A\) and \(B\). Given that \(C\) lies on \(l\),
2. find the value of \(p\) and the value of \(q\),
3. calculate, in degrees, the acute angle between \(O C\) and \(A B\). The point \(D\) lies on \(A B\) and is such that \(O D\) is perpendicular to \(A B\).
4. Find the position vector of \(D\).

4. The points \(A\) and \(B\) have coordinates \(( 3,9 , - 7 )\) and \(( 13 , - 6 , - 2 )\) respectively.

1. Find, in vector form, an equation for the line \(l\) which passes through \(A\) and \(B\).
2. Show that the point \(C\) with coordinates \(( 9,0 , - 4 )\) lies on \(l\). The point \(D\) is the point on \(l\) closest to the origin, \(O\).
3. Find the coordinates of \(D\).
4. Find the area of triangle \(O A B\) to 3 significant figures.
   4. continued

1. A gas company maintains a straight pipeline that passes under a mountain.

The pipeline is modelled as a straight line and one side of the mountain is modelled as a plane. There are accessways from a control centre to two access points on the pipeline.
Modelling the control centre as the origin \(O\), the two access points on the pipeline have coordinates \(P ( - 300,400 , - 150 )\) and \(Q ( 300,300 , - 50 )\), where the units are metres.

1. Find a vector equation for the line \(P Q\), giving your answer in the form \(\mathbf { r } = \mathbf { a } + \lambda \mathbf { b }\), where \(\lambda\) is a scalar parameter. The equation of the plane modelling the side of the mountain is \(2 x + 3 y - 5 z = 300\) The company wants to create a new accessway from this side of the mountain to the pipeline. The accessway will consist of a tunnel of shortest possible length between the pipeline and the point \(M ( 100 , k , 100 )\) on this side of the mountain, where \(k\) is a constant.
2. Using the model, find
   1. the coordinates of the point at which this tunnel will meet the pipeline,
   2. the length of this tunnel. It is only practical to construct the new accessway if it will be significantly shorter than both of the existing accessways, \(O P\) and \(O Q\).
3. Determine whether the company should build the new accessway.
4. Suggest one limitation of the model.

1. An octopus is able to catch any fish that swim within a distance of 2 m from the octopus's position.

A fish \(F\) swims from a point \(A\) to a point \(B\). The octopus is modelled as a fixed particle at the origin \(O\). Fish \(F\) is modelled as a particle moving in a straight line from \(A\) to \(B\). Relative to \(O\), the coordinates of \(A\) are \(( - 3,1 , - 7 )\) and the coordinates of \(B\) are \(( 9,4,11 )\), where the unit of distance is metres.

1. Use the model to determine whether or not the octopus is able to catch fish \(F\).
2. Criticise the model in relation to fish \(F\).
3. Criticise the model in relation to the octopus.

5

1. Find the shortest distance between the point ( \(- 6,4\) ) and the line \(y = - 0.75 x + 7\). Two lines, \(l _ { 1 }\) and \(l _ { 2 }\), are given by \(l _ { 1 } : \mathbf { r } = \left( \begin{array} { c } 4 \\ 3 \\ - 2 \end{array} \right) + \lambda \left( \begin{array} { c } 2 \\ 1 \\ - 4 \end{array} \right)\) and \(l _ { 2 } : \mathbf { r } = \left( \begin{array} { c } 11 \\ - 1 \\ 5 \end{array} \right) + \mu \left( \begin{array} { c } 3 \\ - 1 \\ 1 \end{array} \right)\).
2. Find the shortest distance between \(l _ { 1 }\) and \(l _ { 2 }\).
3. Hence determine the geometrical arrangement of \(l _ { 1 }\) and \(l _ { 2 }\).