4.04h Shortest distances: between parallel lines and between skew lines

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 A mine contains several underground tunnels beneath a hillside. The hillside is a plane, all the tunnels are straight and the width of the tunnels may be neglected. A coordinate system is chosen with the \(z\)-axis pointing vertically upwards and the units are metres. Three points on the hillside have coordinates \(\mathrm { A } ( 15 , - 60,20 )\), \(B ( - 75,100,40 )\) and \(C ( 18,138,35.6 )\).

1. Find the vector product \(\overrightarrow { \mathrm { AB } } \times \overrightarrow { \mathrm { AC } }\) and hence show that the equation of the hillside is \(2 x - 2 y + 25 z = 650\). The tunnel \(T _ { \mathrm { A } }\) begins at A and goes in the direction of the vector \(15 \mathbf { i } + 14 \mathbf { j } - 2 \mathbf { k }\); the tunnel \(T _ { \mathrm { C } }\) begins at C and goes in the direction of the vector \(8 \mathbf { i } + 7 \mathbf { j } - 2 \mathbf { k }\). Both these tunnels extend a long way into the ground.
2. Find the least possible length of a tunnel which connects B to a point in \(T _ { \mathrm { A } }\).
3. Find the least possible length of a tunnel which connects a point in \(T _ { \mathrm { A } }\) to a point in \(T _ { \mathrm { C } }\).
4. A tunnel starts at B , passes through the point ( \(18,138 , p\) ) vertically below C , and intersects \(T _ { \mathrm { A } }\) at the point Q . Find the value of \(p\) and the coordinates of Q .

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 Three points have coordinates \(\mathrm { A } ( - 3,12 , - 7 ) , \mathrm { B } ( - 2,6,9 ) , \mathrm { C } ( 6,0 , - 10 )\). The plane \(P\) passes through the points \(\mathrm { A } , \mathrm { B }\) and C .

1. Find the vector product \(\overrightarrow { \mathrm { AB } } \times \overrightarrow { \mathrm { AC } }\). Hence or otherwise find an equation for the plane \(P\) in the form \(a x + b y + c z = d\). The plane \(Q\) has equation \(6 x + 3 y + 2 z = 32\). The perpendicular from A to the plane \(Q\) meets \(Q\) at the point D. The planes \(P\) and \(Q\) intersect in the line \(L\).
2. Find the distance AD .
3. Find an equation for the line \(L\).
4. Find the shortest distance from A to the line \(L\).
5. Find the volume of the tetrahedron ABCD .

8. 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.

5.The lines \(L _ { 1 }\) and \(L _ { 2 }\) have vector equations \(L _ { 1 } : \quad \mathbf { r } = - 2 \mathbf { i } + 11.5 \mathbf { j } + \lambda ( 3 \mathbf { i } - 4 \mathbf { j } - \mathbf { k } )\), \(L _ { 2 } : \quad \mathbf { r } = 11.5 \mathbf { i } + 3 \mathbf { j } + 8.5 \mathbf { k } + \mu ( 7 \mathbf { i } + 8 \mathbf { j } - 11 \mathbf { k } )\),
where \(\lambda\) and \(\mu\) are parameters.

1. Show that \(L _ { 1 }\) and \(L _ { 2 }\) do not intersect.
2. Show that the vector \(( 2 \mathbf { i } + \mathbf { j } + 2 \mathbf { k } )\) is perpendicular to both \(L _ { 1 }\) and \(L _ { 2 }\) . The point \(A\) lies on \(L _ { 1 }\) ,the point \(B\) lies on \(L _ { 2 }\) and \(A B\) is perpendicular to both \(L _ { 1 }\) and \(L _ { 2 }\) .
3. Find the position vector of the point \(A\) and the position vector of the point \(B\) .
   (8) \includegraphics[max width=\textwidth, alt={}, center]{0df09d8a-7478-4679-b117-128ee226db6a-4_554_1017_404_571} Figure 1 shows a sketch of part of the curve \(C\) with equation $$y = \sin ( \ln x ) , \quad x \geq 1 .$$ The point \(Q\) ,on \(C\) ,is a maximum.

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 .

4 The line \(l\) has equations \(\frac { x - 1 } { 2 } = \frac { y - 1 } { 3 } = \frac { z + 1 } { 2 }\) and the point \(A\) is ( \(7,3,7\) ). \(M\) is the point where the perpendicular from \(A\) meets \(l\).

1. Find, in either order, the coordinates of \(M\) and the perpendicular distance from \(A\) to \(l\).
2. Find the coordinates of the point \(B\) on \(A M\) such that \(\overrightarrow { A B } = 3 \overrightarrow { B M }\).

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

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.

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

1. Find the length of \(P Q\).
2. Find the position vector of \(Q\).
3. Show that the perpendicular distance from \(Q\) to the plane containing \(A B\) and the line \(l _ { 1 }\) is \(\sqrt { } 3\).

6 The line \(l _ { 1 }\) passes through the point with position vector \(8 \mathbf { i } + 8 \mathbf { j } - 7 \mathbf { k }\) and is parallel to the vector \(4 \mathbf { i } + 3 \mathbf { j }\). The line \(l _ { 2 }\) passes through the point with position vector \(7 \mathbf { i } - 2 \mathbf { j } + 4 \mathbf { k }\) and is parallel to the vector \(4 \mathbf { i } - \mathbf { k }\). 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 \(l _ { 2 }\). In either order,

1. show that \(P Q = 13\),
2. find the position vectors of \(P\) and \(Q\).

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 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. }

10 The line \(l _ { 1 }\) is parallel to the vector \(a \mathbf { i } - \mathbf { j } + \mathbf { k }\), where \(a\) is a constant, and passes through the point whose position vector is \(9 \mathbf { j } + 2 \mathbf { k }\). The line \(l _ { 2 }\) is parallel to the vector \(- a \mathbf { i } + 2 \mathbf { j } + 4 \mathbf { k }\) and passes through the point whose position vector is \(- 6 \mathbf { i } - 5 \mathbf { j } + 10 \mathbf { k }\).

1. It is given that \(l _ { 1 }\) and \(l _ { 2 }\) intersect.
   1. Show that \(a = - \frac { 6 } { 13 }\).
   2. Find a cartesian equation of the plane containing \(l _ { 1 }\) and \(l _ { 2 }\).
   3. Given instead that the perpendicular distance between \(l _ { 1 }\) and \(l _ { 2 }\) is \(3 \sqrt { } ( 30 )\), find the value of \(a\).

3 The lines \(l _ { 1 }\) and \(l _ { 2 }\) have equations \(\mathbf { r } = 6 \mathbf { i } + 2 \mathbf { j } + 7 \mathbf { k } + \lambda ( \mathbf { i } + \mathbf { j } )\) and \(\mathbf { r } = 4 \mathbf { i } + 4 \mathbf { j } + \mu ( - 6 \mathbf { j } + \mathbf { k } )\) 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 \(l _ { 2 }\). Find the position vectors of \(P\) and \(Q\).

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 \(D\).

9 With \(O\) as origin, the points \(A , B , C\) have position vectors $$\mathbf { i } , \quad \mathbf { i } + \mathbf { j } , \quad \mathbf { i } + \mathbf { j } + 2 \mathbf { k }$$ respectively. Find a vector equation of the common perpendicular of the lines \(A B\) and \(O C\). Show that the shortest distance between the lines \(A B\) and \(O C\) is \(\frac { 2 } { 5 } \sqrt { } 5\). Find, in the form \(a x + b y + c z = d\), an equation for the plane containing \(A B\) and the common perpendicular of the lines \(A B\) and \(O C\).

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 \(\theta\).

9 Find a cartesian equation of the plane \(\Pi\) containing the lines $$\mathbf { r } = 3 \mathbf { i } + \mathbf { k } + s ( 2 \mathbf { i } + \mathbf { j } - \mathbf { k } ) \quad \text { and } \quad \mathbf { r } = 3 \mathbf { i } - 7 \mathbf { j } + 10 \mathbf { k } + t ( \mathbf { i } - 3 \mathbf { j } + 4 \mathbf { k } )$$ The line \(l\) passes through the point \(P\) with position vector \(6 \mathbf { i } - 2 \mathbf { j } + \mathbf { k }\) and is parallel to the vector \(2 \mathbf { i } + \mathbf { j } - 4 \mathbf { k }\). Find

1. the position vector of the point where \(l\) meets \(\Pi\),
2. the perpendicular distance from \(P\) to \(\Pi\),
3. the acute angle between \(l\) and \(\Pi\).

9 The plane \(\Pi\) has equation $$\mathbf { r } = 2 \mathbf { i } + 3 \mathbf { j } - \mathbf { k } + \lambda ( \mathbf { i } - 2 \mathbf { j } + 2 \mathbf { k } ) + \mu ( 3 \mathbf { i } + \mathbf { j } - 2 \mathbf { k } )$$ The line \(l\), which does not lie in \(\Pi\), has equation $$\mathbf { r } = 3 \mathbf { i } + 6 \mathbf { j } + 12 \mathbf { k } + t ( 8 \mathbf { i } + 5 \mathbf { j } - 8 \mathbf { k } )$$ Show that \(l\) is parallel to \(\Pi\). Find the position vector of the point at which the line with equation \(\mathbf { r } = 5 \mathbf { i } - 4 \mathbf { j } + 7 \mathbf { k } + s ( 2 \mathbf { i } - \mathbf { j } + \mathbf { k } )\) meets \(\Pi\). Find the perpendicular distance from the point with position vector \(9 \mathbf { i } + 11 \mathbf { j } + 2 \mathbf { k }\) to \(l\).