4.04i Shortest distance: between a point and a line

6 The plane with equation \(2 x + 2 y - z = 5\) is denoted by \(m\). Relative to the origin \(O\), the points \(A\) and \(B\) have coordinates \(( 3,4,0 )\) and \(( - 1,0,2 )\) respectively.

1. Show that the plane \(m\) bisects \(A B\) at right angles.
   A second plane \(p\) is parallel to \(m\) and nearer to \(O\). The perpendicular distance between the planes is 1 .
2. Find the equation of \(p\), giving your answer in the form \(a x + b y + c z = d\).

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 \(\Pi\).

7 The lines \(l _ { 1 }\) and \(l _ { 2 }\) have equations \(\mathbf { r } = - 5 \mathbf { j } + \lambda ( 5 \mathbf { i } + 2 \mathbf { k } )\) and \(\mathbf { r } = 4 \mathbf { i } + 2 \mathbf { j } - 2 \mathbf { k } + \mu ( \mathbf { j } + \mathbf { k } )\) respectively. The plane \(\Pi\) contains \(l _ { 1 }\) and is parallel to \(l _ { 2 }\).

1. Find the equation of \(\Pi\), giving your answer in the form \(a x + b y + c z = d\).
2. Find the distance between \(l _ { 2 }\) and \(\Pi\).
   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 \(l _ { 2 }\).
3. Show that \(P\) has position vector \(\frac { 55 } { 27 } \mathbf { i } - 5 \mathbf { j } + \frac { 22 } { 27 } \mathbf { k }\) and state a vector equation for \(P Q\).
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

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.

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

1. Find the equation of the plane \(A B C\), giving your answer in the form \(a x + b y + c z = d\).
2. Find the acute angle between the planes \(O B C\) and \(A B C\).
   The point \(D\) has position vector \(t \mathbf { i } - \mathbf { j }\).
3. Given that the shortest distance between the lines \(A B\) and \(C D\) is \(\sqrt { \mathbf { 1 0 } }\), find the value of \(t\).
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

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

1. Find the length \(P Q\).
   The plane \(\Pi _ { 1 }\) contains \(P Q\) and \(l _ { 1 }\).
   The plane \(\Pi _ { 2 }\) contains \(P Q\) and \(l _ { 2 }\).
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 \(\Pi _ { 2 }\).

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 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 .

1. Show that the only possible value of \(m\) is 2 .
2. Find the shortest distance of \(D\) from the line through \(A\) and \(C\).
3. Show that the acute angle between the planes \(A C D\) and \(B C D\) is \(\cos ^ { - 1 } \left( \frac { 1 } { \sqrt { 3 } } \right)\).

1. With respect to a fixed origin \(O\), the lines \(l _ { 1 }\) and \(l _ { 2 }\) are given by the equations

$$l _ { 1 } : \mathbf { r } = \left( \begin{array} { r } 12 \\ - 4 \\ 5 \end{array} \right) + \lambda \left( \begin{array} { r } 5 \\ - 4 \\ 2 \end{array} \right) , \quad l _ { 2 } : \mathbf { r } = \left( \begin{array} { l } 2 \\ 2 \\ 0 \end{array} \right) + \mu \left( \begin{array} { l } 0 \\ 6 \\ 3 \end{array} \right)$$ 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 \(\left( \begin{array} { l } 7 \\ 0 \\ 3 \end{array} \right)\).
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.

1. The plane \(\Pi\) has equation

$$\mathbf { r } = \left( \begin{array} { l } 1 \\ 2 \\ 3 \end{array} \right) + \lambda \left( \begin{array} { r } 0 \\ 3 \\ - 2 \end{array} \right) + \mu \left( \begin{array} { l } 1 \\ 1 \\ 2 \end{array} \right)$$ where \(\lambda\) and \(\mu\) are scalar parameters.

1. Determine a vector perpendicular to \(\Pi\) The line \(l\) meets \(\Pi\) at the point ( \(1,2,3\) ) and passes through the point ( \(1,0,1\) )
2. Determine the size of the acute angle between \(\Pi\) and \(l\) Give your answer to the nearest degree.
3. Determine the shortest distance between \(\Pi\) and the point \(( 6 , - 3 , - 6 )\)

8. The plane \(\Pi _ { 1 }\) has equation $$x - 5 y + 3 z = 11$$ The plane \(\Pi _ { 2 }\) has equation $$3 x - 2 y + 2 z = 7$$ The planes \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\) intersect in the line \(l\).

1. Find a vector equation for \(l\), giving your answer in the form \(\mathbf { r } = \mathbf { a } + \lambda \mathbf { b }\) where \(\mathbf { a }\) and \(\mathbf { b }\) are constant vectors and \(\lambda\) is a scalar parameter. The point \(P ( 2,0,3 )\) lies on \(\Pi _ { 1 }\) The line \(m\), which passes through \(P\), is parallel to \(l\). The point \(Q ( 3,2,1 )\) lies on \(\Pi _ { 2 }\) The line \(n\), which passes through \(Q\), is also parallel to \(l\).
2. Find, in exact simplified form, the shortest distance between \(m\) and \(n\).
   

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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.

7. The lines \(l _ { 1 }\) and \(l _ { 2 }\) have equations $$\mathbf { r } = \left( \begin{array} { r } 1 \\ - 1 \\ 2 \end{array} \right) + \lambda \left( \begin{array} { r } - 1 \\ 3 \\ 4 \end{array} \right) \text { and } \quad \mathbf { r } = \left( \begin{array} { r } \alpha \\ - 4 \\ 0 \end{array} \right) + \mu \left( \begin{array} { l } 0 \\ 3 \\ 2 \end{array} \right) .$$ If the lines \(l _ { 1 }\) and \(l _ { 2 }\) intersect, find

1. the value of \(\alpha\),
2. an equation for the plane containing the lines \(l _ { 1 }\) and \(l _ { 2 }\), giving your answer in the form \(a x + b y + c z + d = 0\), where \(a , b , c\) and \(d\) are constants. For other values of \(\alpha\), the lines \(l _ { 1 }\) and \(l _ { 2 }\) do not intersect and are skew lines.
   Given that \(\alpha = 2\),
3. find the shortest distance between the lines \(l _ { 1 }\) and \(l _ { 2 }\).

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 A tetrahedron ABCD has vertices \(\mathrm { A } ( - 3,5,2 ) , \mathrm { B } ( 3,13,7 ) , \mathrm { C } ( 7,0,3 )\) and \(\mathrm { D } ( 5,4,8 )\).

1. Find the vector product \(\overrightarrow { \mathrm { AB } } \times \overrightarrow { \mathrm { AC } }\), and hence find the equation of the plane ABC .
2. Find the shortest distance from \(D\) to the plane \(A B C\).
3. Find the shortest distance between the lines AB and CD .
4. Find the volume of the tetrahedron ABCD . The plane \(P\) with equation \(3 x - 2 z + 5 = 0\) contains the point B , and meets the lines AC and AD at E and F respectively.
5. Find \(\lambda\) and \(\mu\) such that \(\overrightarrow { \mathrm { AE } } = \lambda \overrightarrow { \mathrm { AC } }\) and \(\overrightarrow { \mathrm { AF } } = \mu \overrightarrow { \mathrm { AD } }\). Deduce that E is between A and C , and that F is between A and D.
6. Hence, or otherwise, show that \(P\) divides the tetrahedron ABCD into two parts having volumes in the ratio 4 to 17.

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 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.