Shortest distance between two skew lines

5 The lines \(l _ { 1 }\) and \(l _ { 2 }\) have equations \(\mathbf { r } = 3 \mathbf { i } + 3 \mathbf { k } + \lambda ( \mathbf { i } + 4 \mathbf { j } + 4 \mathbf { k } )\) and \(\mathbf { r } = 3 \mathbf { i } - 5 \mathbf { j } - 6 \mathbf { k } + \mu ( 5 \mathbf { j } + 6 \mathbf { k } )\) respectively.

1. Find the shortest distance between \(l _ { 1 }\) and \(l _ { 2 }\).
   The plane \(\Pi\) contains \(l _ { 1 }\) and is parallel to the vector \(\mathbf { i } + \mathbf { k }\).
2. Find the equation of \(\Pi\), giving your answer in the form \(a x + b y + c z = d\).
3. Find the acute angle between \(l _ { 2 }\) and \(\Pi\).

6 Let \(t\) be a positive constant.
The line \(l _ { 1 }\) passes through the point with position vector \(t \mathbf { i } + \mathbf { j }\) and is parallel to the vector \(- 2 \mathbf { i } - \mathbf { j }\). The line \(l _ { 2 }\) passes through the point with position vector \(\mathbf { j } + t \mathbf { k }\) and is parallel to the vector \(- 2 \mathbf { j } + \mathbf { k }\). It is given that the shortest distance between the lines \(l _ { 1 }\) and \(l _ { 2 }\) is \(\sqrt { \mathbf { 2 1 } }\).

1. Find the value of \(t\).
   The plane \(\Pi _ { 1 }\) contains \(l _ { 1 }\) and is parallel to \(l _ { 2 }\).
2. Write down an equation of \(\Pi _ { 1 }\), giving your answer in the form \(\mathbf { r } = \mathbf { a } + \lambda \mathbf { b } + \mu \mathbf { c }\).
   The plane \(\Pi _ { 2 }\) has Cartesian equation \(5 x - 6 y + 7 z = 0\).
3. Find the acute angle between \(l _ { 2 }\) and \(\Pi _ { 2 }\).
4. Find the acute angle between \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\).

7 The position vectors of the points \(A , B , C , D\) are $$7 \mathbf { i } + 4 \mathbf { j } - \mathbf { k } , \quad 11 \mathbf { i } + 3 \mathbf { j } , \quad 2 \mathbf { i } + 6 \mathbf { j } + 3 \mathbf { k } , \quad 2 \mathbf { i } + 7 \mathbf { j } + \lambda \mathbf { k }$$ respectively.

1. Given that the shortest distance between the line \(A B\) and the line \(C D\) is 3 , show that \(\lambda ^ { 2 } - 5 \lambda + 4 = 0\).
   Let \(\Pi _ { 1 }\) be the plane \(A B D\) when \(\lambda = 1\).
   Let \(\Pi _ { 2 }\) be the plane \(A B D\) when \(\lambda = 4\).
2. 1. Write down an equation of \(\Pi _ { 1 }\), giving your answer in the form \(\mathbf { r } = \mathbf { a } + \mathbf { s b } + \mathbf { t c }\).
   2. Find an equation of \(\Pi _ { 2 }\), giving your answer in the form \(a x + b y + c z = d\).
3. Find the acute angle between \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\).
   If you use the following page to complete the answer to any question, the question number must be clearly shown.

5 The lines \(l _ { 1 }\) and \(l _ { 2 }\) have equations \(\mathbf { r } = \mathbf { i } + 4 \mathbf { j } - \mathbf { k } + \lambda ( \mathbf { j } - 2 \mathbf { k } )\) and \(\mathbf { r } = - 3 \mathbf { i } + 4 \mathbf { j } + \mu ( \mathbf { i } + 2 \mathbf { j } + \mathbf { k } )\) respectively.

1. Find the shortest distance between \(l _ { 1 }\) and \(l _ { 2 }\).
   The plane \(\Pi _ { 1 }\) contains \(l _ { 1 }\) and is parallel to \(l _ { 2 }\).
2. Obtain an equation of \(\Pi _ { 1 }\) in the form \(p x + q y + r z = s\). \includegraphics[max width=\textwidth, alt={}, center]{7eb2abb1-68f4-4cc8-8314-f436906d6c4e-10_2715_40_144_2007}
3. The point \(( 1,1,1 )\) lies on the plane \(\Pi _ { 2 }\). It is given that the line of intersection of the planes \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\) passes through the point ( \(0,0,2\) ) and is parallel to the vector \(\mathbf { i } + 4 \mathbf { j } - 3 \mathbf { k }\). Obtain an equation of \(\Pi _ { 2 }\) in the form \(a x + b y + c z = d\).

4 The lines \(l _ { 1 }\) and \(l _ { 2 }\) have equations $$\mathbf { r } = - 2 \mathbf { i } - 3 \mathbf { j } - 5 \mathbf { k } + \lambda ( - 4 \mathbf { i } + 3 \mathbf { j } + 5 \mathbf { k } ) \quad \text { and } \quad \mathbf { r } = 2 \mathbf { i } - 2 \mathbf { j } + 3 \mathbf { k } + \mu ( 2 \mathbf { i } - 3 \mathbf { j } + \mathbf { k } )$$ respectively.

1. Find the shortest distance between \(l _ { 1 }\) and \(l _ { 2 }\).
   The plane \(\Pi\) contains \(l _ { 1 }\) and the point with position vector \(- \mathbf { i } - 3 \mathbf { j } - 4 \mathbf { k }\).
2. Find an equation of \(\Pi\), giving your answer in the form \(a x + b y + c z = d\).

6 Th p itim ctosg th \(\dot { \mathrm { p } }\) ns \(A , B , C , D\) are $$2 \mathbf { i } + 4 \mathbf { j } - 3 \mathbf { k } , \quad - 2 \mathbf { i } + 5 \mathbf { j } - 4 \mathbf { k } , \quad \mathbf { i } + 4 \mathbf { j } + \mathbf { k } , \quad \mathbf { i } + 5 \mathbf { j } + m \mathbf { k }$$ resp ctively, wh re \(m\) is an in eg r. It is g n th the sh test d stan e b tween th lie th g \(\quad A\) ad \(B\) ad lie th g \(\quad C\) ad \(D\) is 3

1. Shat that to b sib ey le \(6 m\) is 2
2. Fid b sh test il stan e \(6 D\) frm th lie thrg \(\mathrm { h } A\) ad \(C\).
3. Swat the actu eas eb tweert b p an \(\mathrm { s } A C D\) ad \(B C D\) is co \({ } ^ { - 1 } \left( \frac { 1 } { \sqrt { 3 } } \right)\).

1. The skew lines \(l _ { 1 }\) and \(l _ { 2 }\) have equations

$$l _ { 1 } : \mathbf { r } = ( \mathbf { i } + 2 \mathbf { j } - 5 \mathbf { k } ) + \lambda ( 5 \mathbf { i } + \mathbf { j } )$$ and $$l _ { 2 } : \mathbf { r } = ( 2 \mathbf { i } - 4 \mathbf { j } + 4 \mathbf { k } ) + \mu ( 8 \mathbf { i } - 2 \mathbf { j } + 3 \mathbf { k } )$$ where \(\lambda\) and \(\mu\) are scalar parameters.

1. Determine a vector that is perpendicular to both \(l _ { 1 }\) and \(l _ { 2 }\)
2. Determine an equation of the plane parallel to \(l _ { 1 }\) that contains \(l _ { 2 }\)
   1. in the form \(\mathbf { r } = \mathbf { a } + s \mathbf { b } + t \mathbf { c }\)
   2. in the form r.n \(= p\)
3. Determine the shortest distance between \(l _ { 1 }\) and \(l _ { 2 }\) Give your answer in simplest form.

2. Two skew lines \(l _ { 1 }\) and \(l _ { 2 }\) have equations $$\begin{aligned} & l _ { 1 } : \mathbf { r } = ( \mathbf { i } - \mathbf { j } + \mathbf { k } ) + \lambda ( 4 \mathbf { i } + 3 \mathbf { j } + 2 \mathbf { k } ) \\ & l _ { 2 } : \mathbf { r } = ( 3 \mathbf { i } + 7 \mathbf { j } + 2 \mathbf { k } ) + \mu ( - 4 \mathbf { i } + 6 \mathbf { j } + \mathbf { k } ) \end{aligned}$$ respectively, where \(\lambda\) and \(\mu\) are real parameters.

1. Find a vector in the direction of the common perpendicular to \(l _ { 1 }\) and \(l _ { 2 }\)
2. Find the shortest distance between these two lines.

6. The line \(l _ { 1 }\) has equation $$\mathbf { r } = \mathbf { i } + 2 \mathbf { k } + \lambda ( 2 \mathbf { i } + 3 \mathbf { j } - \mathbf { k } )$$ where \(\lambda\) is a scalar parameter. The line \(l _ { 2 }\) has equation $$\frac { x + 1 } { 1 } = \frac { y - 4 } { 1 } = \frac { z - 1 } { 3 }$$

1. Prove that the lines \(l _ { 1 }\) and \(l _ { 2 }\) are skew.
2. Find the shortest distance between the lines \(l _ { 1 }\) and \(l _ { 2 }\) The plane \(\Pi\) contains \(l _ { 1 }\) and intersects \(l _ { 2 }\) at the point \(( 3,8,13 )\).
3. Find a cartesian equation for the plane \(\Pi\).

5 The lines \(l _ { 1 }\) and \(l _ { 2 }\) have equations $$\frac { x - 5 } { 1 } = \frac { y - 1 } { - 1 } = \frac { z - 5 } { - 2 } \quad \text { and } \quad \frac { x - 1 } { - 4 } = \frac { y - 11 } { - 14 } = \frac { z - 2 } { 2 } .$$

1. Find the exact value of the shortest distance between \(l _ { 1 }\) and \(l _ { 2 }\).
2. Find an equation for the plane containing \(l _ { 1 }\) and parallel to \(l _ { 2 }\) in the form \(a x + b y + c z = d\).

1 Four points have coordinates \(\mathrm { A } ( - 2 , - 3,2 ) , \mathrm { B } ( - 3,1,5 ) , \mathrm { C } ( k , 5 , - 2 )\) and \(\mathrm { D } ( 0,9 , k )\).

1. Find the vector product \(\overrightarrow { \mathrm { AB } } \times \overrightarrow { \mathrm { CD } }\).
2. For the case when AB is parallel to CD ,
   (A) state the value of \(k\),
   (B) find the shortest distance between the parallel lines AB and CD ,
   (C) find, in the form \(a x + b y + c z + d = 0\), the equation of the plane containing AB and CD .
3. When AB is not parallel to CD , find the shortest distance between the lines AB and CD , in terms of \(k\).
4. Find the value of \(k\) for which the line AB intersects the line CD , and find the coordinates of the point of intersection in this case.

1 Four points have coordinates $$\mathrm { A } ( 3,8,27 ) , \quad \mathrm { B } ( 5,9,25 ) , \quad \mathrm { C } ( 8,0,1 ) \quad \text { and } \quad \mathrm { D } ( 11 , p , p ) ,$$ where \(p\) is a constant.

1. Find the perpendicular distance from C to the line AB .
2. Find \(\overrightarrow { \mathrm { AB } } \times \overrightarrow { \mathrm { CD } }\) in terms of \(p\), and show that the shortest distance between the lines AB and CD is $$\frac { 21 | p - 5 | } { \sqrt { 17 p ^ { 2 } - 2 p + 26 } }$$
3. Find, in terms of \(p\), the volume of the tetrahedron ABCD .
4. State the value of \(p\) for which the lines AB and CD intersect, and find the coordinates of the point of intersection in this case. Option 2: Multi-variable calculus

1 Three points have coordinates \(\mathrm { A } ( 3,2,10 ) , \mathrm { B } ( 11,0 , - 3 ) , \mathrm { C } ( 5,18,0 )\), and \(L\) is the straight line through A with equation $$\frac { x - 3 } { - 1 } = \frac { y - 2 } { 4 } = \frac { z - 10 } { 1 }$$

1. Find the shortest distance between the lines \(L\) and BC .
2. Find the shortest distance from A to the line BC . A straight line passes through B and the point \(\mathrm { P } ( 5,18 , k )\), and intersects the line \(L\).
3. Find \(k\), and the point of intersection of the lines BP and \(L\). The point D is on the line \(L\), and AD has length 12 .
4. Find the volume of the tetrahedron ABCD .

1 The point \(\mathrm { A } ( - 1,12,5 )\) lies on the plane \(P\) with equation \(8 x - 3 y + 10 z = 6\). The point \(\mathrm { B } ( 6 , - 2,9 )\) lies on the plane \(Q\) with equation \(3 x - 4 y - 2 z = 8\). The planes \(P\) and \(Q\) intersect in the line \(L\).

1. Find an equation for the line \(L\).
2. Find the shortest distance between \(L\) and the line AB . The lines \(M\) and \(N\) are both parallel to \(L\), with \(M\) passing through A and \(N\) passing through B .
3. Find the distance between the parallel lines \(M\) and \(N\). The point C has coordinates \(( k , 0,2 )\), and the line AC intersects the line \(N\) at the point D .
4. Find the value of \(k\), and the coordinates of D .

3 Two skew lines have equations $$\frac { x } { 2 } = \frac { y + 3 } { 1 } = \frac { z - 6 } { 3 } \quad \text { and } \quad \frac { x - 5 } { 3 } = \frac { y + 1 } { 1 } = \frac { z - 7 } { 5 } .$$

1. Find the direction of the common perpendicular to the lines.
2. Find the shortest distance between the lines.

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 $$\frac { x - 1 } { 2 } = \frac { y + 2 } { 3 } = \frac { z - 5 } { - 1 } \quad \text { and } \quad \frac { x - 3 } { 4 } = \frac { y - 1 } { - 2 } = \frac { z + 1 } { 3 } .$$

1 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 \(l _ { 2 }\).

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).

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 \(\cos ^ { - 1 } \left( \frac { 1 } { \sqrt { } 3 } \right)\). \footnotetext{Permission to reproduce items where third-party owned material protected by copyright is included has been sought and cleared where possible. Every reasonable effort has been made by the publisher (UCLES) to trace copyright holders, but if any items requiring clearance have unwittingly been included, the publisher will be pleased to make amends at the earliest possible opportunity. University of Cambridge International Examinations is part of the Cambridge Assessment Group. Cambridge Assessment is the brand name of University of Cambridge Local Examinations Syndicate (UCLES), which is itself a department of the University of Cambridge. }

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 \(\cos ^ { - 1 } \left( \frac { 1 } { \sqrt { } 3 } \right)\). \footnotetext{Permission to reproduce items where third-party owned material protected by copyright is included has been sought and cleared where possible. Every reasonable effort has been made by the publisher (UCLES) to trace copyright holders, but if any items requiring clearance have unwittingly been included, the publisher will be pleased to make amends at the earliest possible opportunity. University of Cambridge International Examinations is part of the Cambridge Assessment Group. Cambridge Assessment is the brand name of University of Cambridge Local Examinations Syndicate (UCLES), which is itself a department of the University of Cambridge. }

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 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.
   \footnotetext{Permission to reproduce items where third-party owned material protected by copyright is included has been sought and cleared where possible. Every reasonable effort has been made by the publisher (UCLES) to trace copyright holders, but if any items requiring clearance have unwittingly been included, the publisher will be pleased to make amends at the earliest possible opportunity.
   To avoid the issue of disclosure of answer-related information to candidates, all copyright acknowledgements are reproduced online in the Cambridge International Examinations Copyright Acknowledgements Booklet. This is produced for each series of examinations and is freely available to download at \href{http://www.cie.org.uk}{www.cie.org.uk} after the live examination series. Cambridge International Examinations is part of the Cambridge Assessment Group. Cambridge Assessment is the brand name of University of Cambridge Local Examinations Syndicate (UCLES), which is itself a department of the University of Cambridge. }

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 \(D\).