Shortest distance between two skew lines

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 \(\theta\).

6 With \(O\) as the origin, the points \(A , B , C\) have position vectors $$\mathbf { i } - \mathbf { j } , \quad 2 \mathbf { i } + \mathbf { j } + 7 \mathbf { k } , \quad \mathbf { i } - \mathbf { j } + \mathbf { k }$$ respectively.

1. Find the shortest distance between the lines \(O C\) and \(A B\).
2. Find the cartesian equation of the plane containing the line \(O C\) and the common perpendicular of the lines \(O C\) and \(A B\).

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.

11 A 3-D coordinate system, whose units are metres, is set up to model a construction site. The construction site contains four vertical poles \(P _ { 1 } , P _ { 2 } , P _ { 3 }\) and \(P _ { 4 }\). The floor of the construction site is modelled as lying in the \(x - y\) plane and the poles are modelled as vertical line segments. One end of each pole lies on the floor of the construction site, and the other end of each pole is modelled by the points \(( 0,0,18 ) , ( 12,14,20 ) , ( 0,11,7 )\) and \(( 18,2,16 )\) respectively. A wire, \(S\), runs from the top of \(P _ { 1 }\) to the top of \(P _ { 2 }\). A second wire, \(T\), runs from the top of \(P _ { 3 }\) to the top of \(P _ { 4 }\). The wires are modelled by straight lines segments. The layout of the construction site is illustrated on the diagram below which is not drawn to scale. \includegraphics[max width=\textwidth, alt={}, center]{fbb82fa2-b316-44ae-a19e-197b45f51c87-5_707_871_696_242} A vector equation of the line segment that represents the wire \(S\) is given by \(\mathbf { r } = \left( \begin{array} { c } 0 \\ 0 \\ 18 \end{array} \right) + \lambda \left( \begin{array} { l } 6 \\ 7 \\ 1 \end{array} \right) , 0 \leqslant \lambda \leqslant 2\).

1. Find, in the same form, a vector equation of the line segment that represents the wire \(T\). The components of the direction vector should be integers whose only positive common factor is 1 . For the construction site to be considered safe, it must pass two tests.
   Test 1: The wires \(S\) and \(T\) need to be at least 5 metres apart at all positions on \(S\) and \(T\).
2. By using an appropriate formula, determine whether the construction site passes Test 1. A security camera is placed at a point \(Q\) on wire \(S\). Test 2: To ensure sufficient visibility of the construction site, the distance between the security camera and the top of \(P _ { 3 }\) must be at least 19 m .
3. Determine whether it is possible to find point \(Q\) on \(S\) such that the construction site passes Test 2.

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 \(l _ { 2 }\).

2

1. A plane \(\Pi\) has the equation \(\mathbf { r } \cdot \left( \begin{array} { r } 3 \\ 6 \\ - 2 \end{array} \right) = 15 . C\) is the point \(( 4 , - 5,1 )\).
   Find the shortest distance between \(\Pi\) and \(C\).
2. Lines \(l _ { 1 }\) and \(l _ { 2 }\) have the following equations. $$\begin{aligned} & l _ { 1 } : \mathbf { r } = \left( \begin{array} { l } 4
   3
   1 \end{array} \right)

\end{aligned}

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 The point A has coordinates \(( 2,5,4 )\) and the line BC has equation $$\mathbf { r } = \left( \begin{array} { c } 8 \\ 25 \\ 43 \end{array} \right) + \lambda \left( \begin{array} { c } 4 \\ 15 \\ 25 \end{array} \right)$$ You are given that \(\mathrm { AB } = \mathrm { AC } = 15\).

1. Show that the coordinates of one of the points B and C are (4, 10, 18). Find the coordinates of the other point. These points are B and C respectively.
2. Find the equation of the plane ABC in cartesian form.
3. Show that the plane containing the line BC and perpendicular to the plane ABC has equation \(5 y - 3 z + 4 = 0\). The point D has coordinates \(( 1,1,3 )\).
4. Show that \(| \overrightarrow { B C } \times \overrightarrow { A D } | = \sqrt { 7667 }\) and hence find the shortest distance between the lines \(B C\) and \(A D\).
5. Find the volume of the tetrahedron ABCD .

4 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 \(l _ { 2 }\).

4

1. Find a vector which is perpendicular to both of the vectors $$\mathbf { d } _ { 1 } = \mathbf { i } + 2 \mathbf { j } + 4 \mathbf { k } \quad \text { and } \quad \mathbf { d } _ { 2 } = 9 \mathbf { i } - 3 \mathbf { j } + \mathbf { k } .$$
2. Determine the shortest distance between the skew lines with equations $$\mathbf { r } = 2 \mathbf { i } + 4 \mathbf { j } + 3 \mathbf { k } + \lambda ( \mathbf { i } + 2 \mathbf { j } + 4 \mathbf { k } ) \quad \text { and } \quad \mathbf { r } = \mathbf { i } + \mathbf { j } + 10 \mathbf { k } + \mu ( 9 \mathbf { i } - 3 \mathbf { j } + \mathbf { k } ) .$$

2 Find a vector which is perpendicular to both of the lines $$\mathbf { r } = \left( \begin{array} { r } 11 \\ 5 \\ 4 \end{array} \right) + \lambda \left( \begin{array} { l } 6 \\ 2 \\ 5 \end{array} \right) \quad \text { and } \quad \mathbf { r } = \left( \begin{array} { r } 1 \\ 7 \\ - 1 \end{array} \right) + \mu \left( \begin{array} { r } - 6 \\ 1 \\ 4 \end{array} \right)$$ and hence find the shortest distance between them.

4 Two skew lines have equations \(\mathbf { r } = \left( \begin{array} { r } - 4 \\ 2 \\ 1 \end{array} \right) + \lambda \left( \begin{array} { r } 2 \\ 0 \\ - 1 \end{array} \right)\) and \(\mathbf { r } = \left( \begin{array} { l } 6 \\ 5 \\ 2 \end{array} \right) + \mu \left( \begin{array} { l } 5 \\ 8 \\ 3 \end{array} \right)\). Find a vector which is perpendicular to both lines and determine the shortest distance between the two lines.

11 With respect to an origin \(O\), the points \(A , B , C\) and \(D\) have position vectors $$\mathbf { a } = 2 \mathbf { i } - \mathbf { j } + \mathbf { k } , \quad \mathbf { b } = \mathbf { i } - 2 \mathbf { k } , \quad \mathbf { c } = - \mathbf { i } + 3 \mathbf { j } + 2 \mathbf { k } , \quad \mathbf { d } = - \mathbf { i } + \mathbf { j } + 4 \mathbf { k } ,$$ respectively. Find

1. a vector perpendicular to the plane \(O A B\),
2. the acute angle between the planes \(O A B\) and \(O C D\), correct to the nearest \(0.1 ^ { \circ }\),
3. the shortest distance between the line \(A B\) and the line \(C D\),
4. the perpendicular distance from the point \(A\) to the line \(C D\).

10 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.

The line \(l_1\) passes through the points \((0, 0, 10)\) and \((7, 0, 0)\) and the line \(l_2\) passes through the points \((4, 6, 0)\) and \((3, 3, 1)\). Find the shortest distance between \(l_1\) and \(l_2\). [7]

A theme park has two zip wires. Sarah models the two zip wires as straight lines using coordinates in metres. The ends of one wire are located at \((0, 0, 0)\) and \((0, 100, -20)\) The ends of the other wire are located at \((10, 0, 20)\) and \((-10, 100, -5)\)

1. Use Sarah's model to find the shortest distance between the zip wires. [7 marks]
2. State one way in which Sarah's model could be refined. [1 mark]

1. Find the shortest distance between the point \((-6, 4)\) and the line \(y = -0.75x + 7\). [2]

Two lines, \(l_1\) and \(l_2\), are given by $$l_1: \mathbf{r} = \begin{pmatrix} 4 \\ 3 \\ -2 \end{pmatrix} + \lambda \begin{pmatrix} 2 \\ 1 \\ -4 \end{pmatrix} \text{ and } l_2: \mathbf{r} = \begin{pmatrix} 11 \\ -1 \\ 5 \end{pmatrix} + \mu \begin{pmatrix} 3 \\ -1 \\ 1 \end{pmatrix}$$

1. Find the shortest distance between \(l_1\) and \(l_2\). [3]
2. Hence determine the geometrical arrangement of \(l_1\) and \(l_2\). [2]