Shortest distance from point to line

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.

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 $$l _ { 1 } : \quad \mathbf { r } = \left( \begin{array} { r } 6 \\ - 3 \\ - 2 \end{array} \right) + \lambda \left( \begin{array} { r } - 1 \\ 2 \\ 3 \end{array} \right) , \quad l _ { 2 } : \mathbf { r } = \left( \begin{array} { r } - 5 \\ 15 \\ 3 \end{array} \right) + \mu \left( \begin{array} { r } 2 \\ - 3 \\ 1 \end{array} \right)$$ where \(\lambda\) and \(\mu\) are scalar parameters.

1. Show that \(l _ { 1 }\) and \(l _ { 2 }\) meet and find the position vector of their point of intersection \(A\).
2. Find, to the nearest \(0.1 ^ { \circ }\), the acute angle between \(l _ { 1 }\) and \(l _ { 2 }\). The point \(B\) has position vector \(\left( \begin{array} { r } 5 \\ - 1 \\ 1 \end{array} \right)\).
3. Show that \(B\) lies on \(l _ { 1 }\).
4. Find the shortest distance from \(B\) to the line \(l _ { 2 }\), giving your answer to 3 significant figures. \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \section*{Question 6 continued}

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.

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.

9 The points \(P ( 3,5 , - 21 )\) and \(Q ( - 1,3 , - 16 )\) are on the ceiling of a long straight underground tunnel. A ventilation shaft must be dug from the point \(M\) on the ceiling of the tunnel midway between \(P\) and \(Q\) to horizontal ground level (where the \(z\)-coordinate is 0 ). The ventilation shaft must be perpendicular to the tunnel. The path of the ventilation shaft is modelled by the vector equation \(\mathbf { r } = \mathbf { a } + \lambda \mathbf { b }\), where \(\mathbf { a }\) is the position vector of \(M\). You are given that \(\mathbf { b } = \left( \begin{array} { l } 1 \\ \mathrm {~s} \\ \mathrm { t } \end{array} \right)\) where \(s\) and \(t\) are real numbers.

1. Show that \(\mathrm { S } = 2.5 \mathrm { t } - 2\).
2. Show that at the point where the ventilation shaft reaches the ground \(\lambda = \frac { \mathrm { C } } { \mathrm { t } }\), where \(c\) is a constant to be determined.
3. Using the results in parts (a) and (b), determine the shortest possible length of the ventilation shaft.
4. Explain what the fact that \(\mathbf { b } \times \left( \begin{array} { l } 0 \\ 0 \\ 1 \end{array} \right) \neq \mathbf { O }\) means about the direction of the ventilation shaft.