4.04i Shortest distance: between a point and a line

4 The lines \(l _ { 1 }\) and \(l _ { 2 }\) have equations $$\mathbf { r } = \left( \begin{array} { l } 1 \\ 2 \\ 1 \end{array} \right) + \lambda \left( \begin{array} { r } 2 \\ 3 \\ - 1 \end{array} \right) \text { and } \mathbf { r } = \left( \begin{array} { l } 3 \\ 0 \\ 1 \end{array} \right) + \mu \left( \begin{array} { r } 4 \\ - 1 \\ - 1 \end{array} \right)$$ respectively.

1. Find the shortest distance between the lines.
2. Find a cartesian equation of the plane which contains \(l _ { 1 }\) and which is parallel to \(l _ { 2 }\).

6 Find the shortest distance between the lines with equations $$\frac { x - 1 } { 2 } = \frac { y + 2 } { 3 } = \frac { z - 5 } { - 1 } \quad \text { and } \quad \frac { x - 3 } { 4 } = \frac { y - 1 } { - 2 } = \frac { z + 1 } { 3 } .$$

2 Find the shortest distance between the lines \(\mathbf { r } = \left( \begin{array} { l } 2 \\ 1 \\ 0 \end{array} \right) + \lambda \left( \begin{array} { c } 1 \\ 2 \\ - 1 \end{array} \right)\) and \(\mathbf { r } = \left( \begin{array} { c } - 1 \\ 1 \\ 2 \end{array} \right) + \mu \left( \begin{array} { l } 3 \\ 0 \\ 1 \end{array} \right)\).

The position vectors of the points \(A , B , C , D\) are \(7 \mathbf { i } + 4 \mathbf { j } - \mathbf { k }\), \(3 \mathbf { i } + 5 \mathbf { j } - 2 \mathbf { k }\), \(2 \mathbf { i } + 6 \mathbf { j } + 3 \mathbf { k }\), \(2 \mathbf { i } + 7 \mathbf { j } + \lambda \mathbf { k }\) respectively. It is given that the shortest distance between the line \(A B\) and the line \(C D\) is 3 .

1. Show that \(\lambda ^ { 2 } - 5 \lambda + 4 = 0\).
2. Find the acute angle between the planes through \(A , B , D\) corresponding to the values of \(\lambda\) satisfying the equation in part (i).

11 The line \(l _ { 1 }\) is parallel to the vector \(4 \mathbf { j } - \mathbf { k }\) and passes through the point \(A\) whose position vector is \(2 \mathbf { i } + \mathbf { j } + 4 \mathbf { k }\). The variable line \(l _ { 2 }\) is parallel to the vector \(\mathbf { i } - ( 2 \sin t ) \mathbf { j }\), where \(0 \leqslant t < 2 \pi\), and passes through the point \(B\) whose position vector is \(\mathbf { i } + 2 \mathbf { j } + 4 \mathbf { k }\). The points \(P\) and \(Q\) are on \(l _ { 1 }\) and \(l _ { 2 }\), respectively, and \(P Q\) is perpendicular to both \(l _ { 1 }\) and \(l _ { 2 }\).

1. Find the length of \(P Q\) in terms of \(t\).
2. Hence find the values of \(t\) for which \(l _ { 1 }\) and \(l _ { 2 }\) intersect.
3. For the case \(t = \frac { 1 } { 4 } \pi\), find the perpendicular distance from \(A\) to the plane \(B P Q\), giving your answer correct to 3 decimal places.

10 The lines \(l _ { 1 }\) and \(l _ { 2 }\) have equations $$l _ { 1 } : \mathbf { r } = 6 \mathbf { i } + 5 \mathbf { j } + 4 \mathbf { k } + \lambda ( \mathbf { i } + \mathbf { j } + \mathbf { k } ) \quad \text { and } \quad l _ { 2 } : \mathbf { r } = 6 \mathbf { i } + 5 \mathbf { j } + 4 \mathbf { k } + \mu ( 4 \mathbf { i } + 6 \mathbf { j } + \mathbf { k } ) .$$ Find a cartesian equation of the plane \(\Pi\) containing \(l _ { 1 }\) and \(l _ { 2 }\). Find the position vector of the foot of the perpendicular from the point with position vector \(\mathbf { i } + 10 \mathbf { j } + 3 \mathbf { k }\) to \(\Pi\). The line \(l _ { 3 }\) has equation \(\mathbf { r } = \mathbf { i } + 10 \mathbf { j } + 3 \mathbf { k } + v ( 2 \mathbf { i } - 3 \mathbf { j } + \mathbf { k } )\). Find the shortest distance between \(l _ { 1 }\) and \(l _ { 3 }\).

The position vectors of the points \(A , B , C , D\) are $$2 \mathbf { i } + 4 \mathbf { j } - 3 \mathbf { k } , \quad - 2 \mathbf { i } + 5 \mathbf { j } - 4 \mathbf { k } , \quad \mathbf { i } + 4 \mathbf { j } + \mathbf { k } , \quad \mathbf { i } + 5 \mathbf { j } + m \mathbf { k }$$ respectively, where \(m\) is an integer. It is given that the shortest distance between the line through \(A\) and \(B\) and the line through \(C\) and \(D\) is 3 . Show that the only possible value of \(m\) is 2 . Find the shortest distance of \(D\) from the line through \(A\) and \(C\). Show that the acute angle between the planes \(A C D\) and \(B C D\) is \(\cos ^ { - 1 } \left( \frac { 1 } { \sqrt { } 3 } \right)\).

The position vectors of the points \(A , B , C , D\) are $$2 \mathbf { i } + 4 \mathbf { j } - 3 \mathbf { k } , \quad - 2 \mathbf { i } + 5 \mathbf { j } - 4 \mathbf { k } , \quad \mathbf { i } + 4 \mathbf { j } + \mathbf { k } , \quad \mathbf { i } + 5 \mathbf { j } + m \mathbf { k } ,$$ respectively, where \(m\) is an integer. It is given that the shortest distance between the line through \(A\) and \(B\) and the line through \(C\) and \(D\) is 3 . Show that the only possible value of \(m\) is 2 . Find the shortest distance of \(D\) from the line through \(A\) and \(C\). Show that the acute angle between the planes \(A C D\) and \(B C D\) is \(\cos ^ { - 1 } \left( \frac { 1 } { \sqrt { } 3 } \right)\).

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.

11 The line \(l _ { 1 }\) passes through the points \(A ( 2,3 , - 5 )\) and \(B ( 8,7 , - 13 )\). The line \(l _ { 2 }\) passes through the points \(C ( - 2,1,8 )\) and \(D ( 3 , - 1,4 )\). Find the shortest distance between the lines \(l _ { 1 }\) and \(l _ { 2 }\). The plane \(\Pi _ { 1 }\) passes through the points \(A , B\) and \(D\). The plane \(\Pi _ { 2 }\) passes though the points \(A , C\) and \(D\). Find the acute angle between \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\), giving your answer in degrees.

The lines \(l _ { 1 }\) and \(l _ { 2 }\) have equations \(\mathbf { r } = 8 \mathbf { i } + 2 \mathbf { j } + 3 \mathbf { k } + \lambda ( \mathbf { i } - 2 \mathbf { j } )\) and \(\mathbf { r } = 5 \mathbf { i } + 3 \mathbf { j } - 14 \mathbf { k } + \mu ( 2 \mathbf { j } - 3 \mathbf { k } )\) respectively. The point \(P\) on \(l _ { 1 }\) and the point \(Q\) on \(l _ { 2 }\) are such that \(P Q\) is perpendicular to both \(l _ { 1 }\) and \(l _ { 2 }\). Find the position vector of the point \(P\) and the position vector of the point \(Q\). The points with position vectors \(8 \mathbf { i } + 2 \mathbf { j } + 3 \mathbf { k }\) and \(5 \mathbf { i } + 3 \mathbf { j } - 14 \mathbf { k }\) are denoted by \(A\) and \(B\) respectively. Find

1. \(\overrightarrow { A P } \times \overrightarrow { A Q }\) and hence the area of the triangle \(A P Q\),
2. the volume of the tetrahedron \(A P Q B\). (You are given that the volume of a tetrahedron is \(\frac { 1 } { 3 } \times\) area of base × perpendicular height.) {www.cie.org.uk} after the live examination series.
   }

The position vectors of the points \(A , B , C , D\) are $$\mathbf { i } + \mathbf { j } + 3 \mathbf { k } , \quad 3 \mathbf { i } - \mathbf { j } + 5 \mathbf { k } , \quad 3 \mathbf { i } - \mathbf { j } + \mathbf { k } , \quad 5 \mathbf { i } - 5 \mathbf { j } + \alpha \mathbf { k } ,$$ respectively, where \(\alpha\) is a positive integer. It is given that the shortest distance between the line \(A B\) and the line \(C D\) is equal to \(2 \sqrt { } 2\).

1. Show that the possible values of \(\alpha\) are 3 and 5 .
2. Using \(\alpha = 3\), find the shortest distance of the point \(D\) from the line \(A C\), giving your answer correct to 3 significant figures.
3. Using \(\alpha = 3\), find the acute angle between the planes \(A B C\) and \(A B D\), giving your answer in degrees.
   {www.cie.org.uk} after the live examination series. }

7 The line \(l _ { 1 }\) passes through the points \(A ( - 3,1,4 )\) and \(B ( - 1,5,9 )\). The line \(l _ { 2 }\) passes through the points \(C ( - 2,6,5 )\) and \(D ( - 1,7,5 )\).

1. Find the shortest distance between the lines \(l _ { 1 }\) and \(l _ { 2 }\).
2. Find the acute angle between the line \(l _ { 2 }\) and the plane containing \(A , B\) and \(D\).

2 Relative to an origin \(O\), the points \(A , B , C\) have position vectors $$\mathbf { i } , \quad \mathbf { j } + \mathbf { k } , \quad \mathbf { i } + \mathbf { j } + \theta \mathbf { k }$$ respectively. The shortest distance between the lines \(A B\) and \(O C\) is \(\frac { 1 } { \sqrt { 2 } }\). Find the value of \(\theta\).

10 The line \(l _ { 1 }\) is parallel to the vector \(\mathbf { i } - 2 \mathbf { j } - 3 \mathbf { k }\) and passes through the point \(A\), whose position vector is \(3 \mathbf { i } + 3 \mathbf { j } - 4 \mathbf { k }\). The line \(l _ { 2 }\) is parallel to the vector \(- 2 \mathbf { i } + \mathbf { j } + 3 \mathbf { k }\) and passes through the point \(B\), whose position vector is \(- 3 \mathbf { i } - \mathbf { j } + 2 \mathbf { k }\). The point \(P\) on \(l _ { 1 }\) and the point \(Q\) on \(l _ { 2 }\) are such that \(P Q\) is perpendicular to both \(l _ { 1 }\) and \(l _ { 2 }\). Find

1. the length \(P Q\),
2. the cartesian equation of the plane \(\Pi\) containing \(P Q\) and \(l _ { 2 }\),
3. the perpendicular distance of \(A\) from \(\Pi\).

The lines \(l _ { 1 }\) and \(l _ { 2 }\) have equations $$\mathbf { r } = 6 \mathbf { i } - 3 \mathbf { j } + s ( 3 \mathbf { i } - 4 \mathbf { j } - 2 \mathbf { k } ) \quad \text { and } \quad \mathbf { r } = 2 \mathbf { i } - \mathbf { j } - 4 \mathbf { k } + t ( \mathbf { i } - 3 \mathbf { j } - \mathbf { k } )$$ respectively. The point \(P\) on \(l _ { 1 }\) and the point \(Q\) on \(l _ { 2 }\) are such that \(P Q\) is perpendicular to both \(l _ { 1 }\) and \(l _ { 2 }\). Show that the position vector of \(P\) is \(3 \mathbf { i } + \mathbf { j } + 2 \mathbf { k }\) and find the position vector of \(Q\). Find, in the form \(\mathbf { r } = \mathbf { a } + \lambda \mathbf { b } + \mu \mathbf { c }\), an equation of the plane \(\Pi\) which passes through \(P\) and is perpendicular to \(l _ { 1 }\). The plane \(\Pi\) meets the plane \(\mathbf { r } = p \mathbf { i } + q \mathbf { j }\) in the line \(l _ { 3 }\). Find a vector equation of \(l _ { 3 }\).

1 The points \(\mathrm { A } ( 2 , - 1,3 ) , \mathrm { B } ( - 2 , - 7,7 )\) and \(\mathrm { C } ( 7,5,1 )\) are three vertices of a tetrahedron ABCD .
The plane ABD has equation \(x + 4 y + 7 z = 19\).
The plane ACD has equation \(2 x - y + 2 z = 11\).

1. Find the shortest distance from \(B\) to the plane \(A C D\).
2. Find an equation for the line AD .
3. Find the shortest distance from C to the line AD .
4. Find the shortest distance between the lines \(A D\) and \(B C\).
5. Given that the tetrahedron ABCD has volume 20, find the coordinates of the two possible positions for the vertex \(D\).

1 Three planes \(P , Q\) and \(R\) have the following equations. $$\begin{array} { l l } \text { Plane } P : & 8 x - y - 14 z = 20 \\ \text { Plane } Q : & 6 x + 2 y - 5 z = 26 \\ \text { Plane } R : & 2 x + y - z = 40 \end{array}$$ The line of intersection of the planes \(P\) and \(Q\) is \(K\).
The line of intersection of the planes \(P\) and \(R\) is \(L\).

1. Show that \(K\) and \(L\) are parallel lines, and find the shortest distance between them.
2. Show that the shortest distance between the line \(K\) and the plane \(R\) is \(5 \sqrt { 6 }\). The line \(M\) has equation \(\mathbf { r } = ( \mathbf { i } - 4 \mathbf { j } ) + \lambda ( 5 \mathbf { i } - 4 \mathbf { j } + 3 \mathbf { k } )\).
3. Show that the lines \(K\) and \(M\) intersect, and find the coordinates of the point of intersection.
4. Find the shortest distance between the lines \(L\) and \(M\).

1 Positions in space around an aerodrome are modelled by a coordinate system with a point on the runway as the origin, O . The \(x\)-axis is east, the \(y\)-axis is north and the \(z\)-axis is vertically upwards. Units of distance are kilometres. Units of time are hours.
At time \(t = 0\), an aeroplane, P , is at \(( 3,4,8 )\) and is travelling in a direction \(\left( \begin{array} { l } 2 \\ 1 \\ 0 \end{array} \right)\) at a constant speed of \(900 \mathrm { kmh } ^ { - 1 }\).

1. Find the least distance of the path of P from the point O . At time \(t = 0\), a second aeroplane, Q , is at \(( 80,40,10 )\). It is travelling in a straight line towards the point O . Its speed is constant at \(270 \mathrm { kmh } ^ { - 1 }\).
2. Show that the shortest distance between the paths of the two aeroplanes is 2.24 km correct to three significant figures.
3. By finding the points on the paths where the shortest distance occurs and the times at which the aeroplanes are at these points, show that in fact the aeroplanes are never this close.
4. A third aeroplane, R , is at position \(( 29,19,5.5 )\) at time \(t = 0\) and is travelling at \(285 \mathrm {~km} \mathrm {~h} ^ { - 1 }\) in a direction \(\left( \begin{array} { c } 18 \\ 6 \\ 1 \end{array} \right)\). Given that Q is in the process of landing and cannot change course, show that R needs to be instructed to alter course or change speed.

6 The equations of two non-intersecting lines, \(l _ { 1 }\) and \(l _ { 2 }\), are \(l _ { 1 } : \mathbf { r } = \left( \begin{array} { c } 1 \\ 2 \\ - 1 \end{array} \right) + \lambda \left( \begin{array} { c } 2 \\ 1 \\ - 2 \end{array} \right) , \quad l _ { 2 } : \mathbf { r } = \left( \begin{array} { c } 2 \\ 2 \\ - 3 \end{array} \right) + \mu \left( \begin{array} { c } 1 \\ - 1 \\ 4 \end{array} \right)\).
Find the shortest distance between lines \(l _ { 1 }\) and \(l _ { 2 }\).

7 The lines \(l _ { 1 }\) and \(l _ { 2 }\) have equations \(\frac { x - 3 } { 1 } = \frac { y - 5 } { 2 } = \frac { z + 2 } { - 3 }\) and \(\frac { x - 4 } { 2 } = \frac { y + 2 } { - 1 } = \frac { z - 7 } { 4 }\).

1. Find the shortest distance between \(l _ { 1 }\) and \(l _ { 2 }\).
2. Find a cartesian equation of the plane which contains \(l _ { 1 }\) and is parallel to \(l _ { 2 }\).

5
2
4 \end{array} \right) + \mu \left( \begin{array} { r } 1
- 2
1 \end{array} \right) $$ Find, in exact form, the distance between \(l _ { 1 }\) and \(l _ { 2 }\).

11

1. Given that \(\mathbf { u } = \lambda \mathbf { i } + \mathbf { j } - 3 \mathbf { k }\) and \(\mathbf { v } = \mathbf { i } + 2 \mathbf { j } - 2 \mathbf { k }\), find the following, giving your answers in terms of \(\lambda\).
   1. u.v
   2. \(\mathbf { u } \times \mathbf { v }\)
2. Hence determine
   1. the acute angle between the planes \(2 x + y - 3 z = 10\) and \(x + 2 y - 2 z = 10\),
   2. the shortest distance between the lines \(\frac { x - 3 } { 3 } = \frac { y } { 1 } = \frac { z - 2 } { - 3 }\) and \(\frac { x } { 1 } = \frac { y - 4 } { 2 } = \frac { z + 2 } { - 2 }\), giving your answer as a multiple of \(\sqrt { 2 }\).

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.