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

8 The point \(P\) has coordinates \(( - 1,4,11 )\) and the line \(l\) has equation \(\mathbf { r } = \left( \begin{array} { r } 1 \\ 3 \\ - 4 \end{array} \right) + \lambda \left( \begin{array} { l } 2 \\ 1 \\ 3 \end{array} \right)\).

1. Find the perpendicular distance from \(P\) to \(l\).
2. Find the equation of the plane which contains \(P\) and \(l\), giving your answer in the form \(a x + b y + c z = d\), where \(a , b , c\) and \(d\) are integers.

7 The straight line \(l\) has equation \(\mathbf { r } = 4 \mathbf { i } - \mathbf { j } + 2 \mathbf { k } + \lambda ( 2 \mathbf { i } - 3 \mathbf { j } + 6 \mathbf { k } )\). The plane \(p\) passes through the point \(( 4 , - 1,2 )\) and is perpendicular to \(l\).

1. Find the equation of \(p\), giving your answer in the form \(a x + b y + c z = d\).
2. Find the perpendicular distance from the origin to \(p\).
3. A second plane \(q\) is parallel to \(p\) and the perpendicular distance between \(p\) and \(q\) is 14 units. Find the possible equations of \(q\).

8 The points \(A\) and \(B\) have position vectors, relative to the origin \(O\), given by \(\overrightarrow { O A } = \mathbf { i } + \mathbf { j } + \mathbf { k }\) and \(\overrightarrow { O B } = 2 \mathbf { i } + 3 \mathbf { k }\). The line \(l\) has vector equation \(\mathbf { r } = 2 \mathbf { i } - 2 \mathbf { j } - \mathbf { k } + \mu ( - \mathbf { i } + 2 \mathbf { j } + \mathbf { k } )\).

1. Show that the line passing through \(A\) and \(B\) does not intersect \(l\).
2. Show that the length of the perpendicular from \(A\) to \(l\) is \(\frac { 1 } { \sqrt { 2 } }\).

10 The line \(l\) has equation \(\mathbf { r } = \mathbf { i } + 2 \mathbf { j } + 3 \mathbf { k } + \mu ( 2 \mathbf { i } - \mathbf { j } - 2 \mathbf { k } )\).

1. The point \(P\) has position vector \(4 \mathbf { i } + 2 \mathbf { j } - 3 \mathbf { k }\). Find the length of the perpendicular from \(P\) to \(l\).
2. It is given that \(l\) lies in the plane with equation \(a x + b y + 2 z = 13\), where \(a\) and \(b\) are constants. Find the values of \(a\) and \(b\).
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

10 With respect to the origin \(O\), the points \(A , B\) and \(C\) have position vectors given by $$\overrightarrow { O A } = \left( \begin{array} { r } 3 \\ - 2 \\ 4 \end{array} \right) , \quad \overrightarrow { O B } = \left( \begin{array} { r } 2 \\ - 1 \\ 7 \end{array} \right) \quad \text { and } \quad \overrightarrow { O C } = \left( \begin{array} { r } 1 \\ - 5 \\ - 3 \end{array} \right) .$$ The plane \(m\) is parallel to \(\overrightarrow { O C }\) and contains \(A\) and \(B\).

1. Find the equation of \(m\), giving your answer in the form \(a x + b y + c z = d\).
2. Find the length of the perpendicular from \(C\) to the line through \(A\) and \(B\).

10 With respect to the origin \(O\), the points \(A , B\) and \(C\) have position vectors given by $$\overrightarrow { O A } = \left( \begin{array} { r } 3 \\ - 2 \\ 4 \end{array} \right) , \quad \overrightarrow { O B } = \left( \begin{array} { r } 2 \\ - 1 \\ 7 \end{array} \right) \quad \text { and } \quad \overrightarrow { O C } = \left( \begin{array} { r } 1 \\ - 5 \\ - 3 \end{array} \right)$$ The plane \(m\) is parallel to \(\overrightarrow { O C }\) and contains \(A\) and \(B\).

1. Find the equation of \(m\), giving your answer in the form \(a x + b y + c z = d\).
2. Find the length of the perpendicular from \(C\) to the line through \(A\) and \(B\).

10 The line \(l\) has equation \(\mathbf { r } = 4 \mathbf { i } - 9 \mathbf { j } + 9 \mathbf { k } + \lambda ( - 2 \mathbf { i } + \mathbf { j } - 2 \mathbf { k } )\). The point \(A\) has position vector \(3 \mathbf { i } + 8 \mathbf { j } + 5 \mathbf { k }\).

1. Show that the length of the perpendicular from \(A\) to \(l\) is 15 .
2. The line \(l\) lies in the plane with equation \(a x + b y - 3 z + 1 = 0\), where \(a\) and \(b\) are constants. Find the values of \(a\) and \(b\).

7 The plane \(m\) has equation \(x + 4 y - 8 z = 2\). The plane \(n\) is parallel to \(m\) and passes through the point \(P\) with coordinates \(( 5,2 , - 2 )\).

1. Find the equation of \(n\), giving your answer in the form \(a x + b y + c z = d\).
2. Calculate the perpendicular distance between \(m\) and \(n\).
3. The line \(l\) lies in the plane \(n\), passes through the point \(P\) and is perpendicular to \(O P\), where \(O\) is the origin. Find a vector equation for \(l\).

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.

5 The points \(A , B , C\) have position vectors $$2 \mathbf { i } + 2 \mathbf { j } + 4 \mathbf { k } , \quad 2 \mathbf { i } + 4 \mathbf { j } - \mathbf { k } , \quad - 3 \mathbf { i } - 3 \mathbf { j } + 4 \mathbf { k }$$ respectively, relative to the origin \(O\).

1. Find the equation of the plane \(A B C\), giving your answer in the form \(a x + b y + c z = d\).
   The point \(D\) has position vector \(2 \mathbf { i } + \mathbf { j } + 3 \mathbf { k }\).
2. Find the perpendicular distance from \(D\) to the plane \(A B C\).
3. Find the shortest distance between the lines \(A B\) and \(C D\).

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 \(a x + b y + c z = d\).

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.

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 \(a x + b y + c z = d\).

9 Let \(\mathrm { f } ( x ) = \frac { 2 + 11 x - 10 x ^ { 2 } } { ( 1 + 2 x ) ( 1 - 2 x ) ( 2 + x ) }\).

1. Express \(\mathrm { f } ( x )\) in partial fractions.
2. Hence obtain the expansion of \(\mathrm { f } ( x )\) in ascending powers of \(x\), up to and including the term in \(x ^ { 2 }\).

6. With respect to a fixed origin \(O\), the lines \(l _ { 1 }\) and \(l _ { 2 }\) are given by the equations $$l _ { 1 } : \quad \mathbf { r } = \left( \begin{array} { r } 6 \\ - 3 \\ - 2 \end{array} \right) + \lambda \left( \begin{array} { r } - 1 \\ 2 \\ 3 \end{array} \right) , \quad l _ { 2 } : \mathbf { r } = \left( \begin{array} { r } - 5 \\ 15 \\ 3 \end{array} \right) + \mu \left( \begin{array} { r } 2 \\ - 3 \\ 1 \end{array} \right)$$ 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 \(\left( \begin{array} { r } 5 \\ - 1 \\ 1 \end{array} \right)\).
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. \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \section*{Question 6 continued}

7. With respect to a fixed origin \(O\),

• the line \(l\) has equation \(\mathbf { r } = \left( \begin{array} { r } 4 \\ 2 \\ - 3 \end{array} \right) + \lambda \left( \begin{array} { r } - 4 \\ - 3 \\ 5 \end{array} \right)\) where \(\lambda\) is a scalar constant
• the point \(A\) has position vector \(9 \mathbf { i } - 3 \mathbf { j } + 2 \mathbf { k }\)

Given that \(X\) is the point on \(l\) nearest to \(A\),

1. find
   1. the coordinates of \(X\)
   2. the shortest distance from \(A\) to \(l\). Give your answer in the form \(\sqrt { d }\), where \(d\) is an integer. The point \(B\) is the image of \(A\) after reflection in \(l\).
2. Find the position vector of \(B\). Solutions relying on calculator technology are not acceptable. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{08756c4b-6619-42da-ac8a-2bf065c01de8-26_668_661_408_644} \captionsetup{labelformat=empty} \caption{Figure 3}
   \end{figure}

9 Two lines \(l\) and \(m\) have equations \(\mathbf { r } = 3 \mathbf { i } + 2 \mathbf { j } + 5 \mathbf { k } + s ( 4 \mathbf { i } - \mathbf { j } + 3 \mathbf { k } )\) and \(\mathbf { r } = \mathbf { i } - \mathbf { j } - 2 \mathbf { k } + t ( - \mathbf { i } + 2 \mathbf { j } + 2 \mathbf { k } )\) respectively.

1. Show that \(l\) and \(m\) are perpendicular.
2. Show that \(l\) and \(m\) intersect and state the position vector of the point of intersection.
3. Show that the length of the perpendicular from the origin to the line \(m\) is \(\frac { 1 } { 3 } \sqrt { 5 }\).

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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7. The plane \(\Pi\) has vector equation $$\mathbf { r } = 3 \mathbf { i } + \mathbf { k } + \lambda ( - 4 \mathbf { i } + \mathbf { j } ) + \mu ( 6 \mathbf { i } - 2 \mathbf { j } + \mathbf { k } )$$

1. Find an equation of \(\Pi\) in the form \(\mathbf { r } \cdot \mathbf { n } = p\), where \(\mathbf { n }\) is a vector perpendicular to \(\Pi\) and \(p\) is a constant. The point \(P\) has coordinates \(( 6,13,5 )\). The line \(l\) passes through \(P\) and is perpendicular to \(\Pi\). The line \(l\) intersects \(\Pi\) at the point \(N\).
2. Show that the coordinates of \(N\) are \(( 3,1 , - 1 )\). The point \(R\) lies on \(\Pi\) and has coordinates \(( 1,0,2 )\).
3. Find the perpendicular distance from \(N\) to the line \(P R\). Give your answer to 3 significant figures.

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.