Line intersection with line

10 The lines \(l\) and \(m\) have vector equations $$\mathbf { r } = \mathbf { i } + \mathbf { j } + \mathbf { k } + s ( \mathbf { i } - \mathbf { j } + 2 \mathbf { k } ) \quad \text { and } \quad \mathbf { r } = 4 \mathbf { i } + 6 \mathbf { j } + \mathbf { k } + t ( 2 \mathbf { i } + 2 \mathbf { j } + \mathbf { k } )$$ respectively.

1. Show that \(l\) and \(m\) intersect.
2. Calculate the acute angle between the lines.
3. Find the equation of the plane containing \(l\) and \(m\), giving your answer in the form \(a x + b y + c z = d\). \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. }

10 With respect to the origin \(O\), the lines \(l\) and \(m\) have vector 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.

1. Prove that \(l\) and \(m\) do not intersect.
2. Calculate the acute angle between the directions of \(l\) and \(m\).
3. Find the equation of the plane which is parallel to \(l\) and contains \(m\), giving your answer in the form \(a x + b y + c z = d\).

10 The lines \(l\) and \(m\) have vector equations $$\mathbf { r } = \mathbf { i } - 2 \mathbf { k } + s ( 2 \mathbf { i } + \mathbf { j } + 3 \mathbf { k } ) \quad \text { and } \quad \mathbf { r } = 6 \mathbf { i } - 5 \mathbf { j } + 4 \mathbf { k } + t ( \mathbf { i } - 2 \mathbf { j } + \mathbf { k } )$$ respectively.

1. Show that \(l\) and \(m\) intersect, and find the position vector of their point of intersection.
2. Find the equation of the plane containing \(l\) and \(m\), giving your answer in the form \(a x + b y + c z = d\).

6 With respect to the origin \(O\), the points \(A , B\) and \(C\) have position vectors given by $$\overrightarrow { O A } = \mathbf { i } - \mathbf { k } , \quad \overrightarrow { O B } = 3 \mathbf { i } + 2 \mathbf { j } - 3 \mathbf { k } \quad \text { and } \quad \overrightarrow { O C } = 4 \mathbf { i } - 3 \mathbf { j } + 2 \mathbf { k }$$ The mid-point of \(A B\) is \(M\). The point \(N\) lies on \(A C\) between \(A\) and \(C\) and is such that \(A N = 2 N C\).

1. Find a vector equation of the line \(M N\).
2. It is given that \(M N\) intersects \(B C\) at the point \(P\). Find the position vector of \(P\).

8 Two lines have equations $$\mathbf { r } = \left( \begin{array} { r } 5 \\ 1 \\ - 4 \end{array} \right) + s \left( \begin{array} { r } 1 \\ - 1 \\ 3 \end{array} \right) \quad \text { and } \quad \mathbf { r } = \left( \begin{array} { r } p \\ 4 \\ - 2 \end{array} \right) + t \left( \begin{array} { r } 2 \\ 5 \\ - 4 \end{array} \right) ,$$ where \(p\) is a constant. It is given that the lines intersect.

1. Find the value of \(p\) and determine the coordinates of the point of intersection.
2. Find the equation of the plane containing the two lines, giving your answer in the form \(a x + b y + c z = d\), where \(a , b , c\) and \(d\) are integers.

7 The equations of two straight lines are $$\mathbf { r } = \mathbf { i } + 4 \mathbf { j } - 2 \mathbf { k } + \lambda ( \mathbf { i } + 3 \mathbf { k } ) \quad \text { and } \quad \mathbf { r } = a \mathbf { i } + 2 \mathbf { j } - 2 \mathbf { k } + \mu ( \mathbf { i } + 2 \mathbf { j } + 3 a \mathbf { k } )$$ where \(a\) is a constant.

1. Show that the lines intersect for all values of \(a\).
2. Given that the point of intersection is at a distance of 9 units from the origin, find the possible values of \(a\).

10 With respect to the origin \(O\), the points \(A\) and \(B\) have position vectors given by \(\overrightarrow { O A } = 6 \mathbf { i } + 2 \mathbf { j }\) and \(\overrightarrow { O B } = 2 \mathbf { i } + 2 \mathbf { j } + 3 \mathbf { k }\). The midpoint of \(O A\) is \(M\). The point \(N\) lying on \(A B\), between \(A\) and \(B\), is such that \(A N = 2 N B\).

1. Find a vector equation for the line through \(M\) and \(N\).
   The line through \(M\) and \(N\) intersects the line through \(O\) and \(B\) at the point \(P\).
2. Find the position vector of \(P\).
3. Calculate angle \(O P M\), giving your answer in degrees.
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

9 The lines \(l\) and \(m\) have vector equations $$\mathbf { r } = - \mathbf { i } + 3 \mathbf { j } + 4 \mathbf { k } + \lambda ( 2 \mathbf { i } - \mathbf { j } - \mathbf { k } ) \quad \text { and } \quad \mathbf { r } = 5 \mathbf { i } + 4 \mathbf { j } + 3 \mathbf { k } + \mu ( a \mathbf { i } + b \mathbf { j } + \mathbf { k } )$$ respectively, where \(a\) and \(b\) are constants.

1. Given that \(l\) and \(m\) intersect, show that \(2 b - a = 4\).
2. Given also that \(l\) and \(m\) are perpendicular, find the values of \(a\) and \(b\).
3. When \(a\) and \(b\) have these values, find the position vector of the point of intersection of \(l\) and \(m\).

8 The points \(A , B\) and \(C\) have position vectors \(\overrightarrow { \mathrm { OA } } = - 2 \mathbf { i } + \mathbf { j } + 4 \mathbf { k } , \overrightarrow { \mathrm { OB } } = 5 \mathbf { i } + 2 \mathbf { j }\) and \(\overrightarrow { \mathrm { OC } } = 8 \mathbf { i } + 5 \mathbf { j } - 3 \mathbf { k }\), where \(O\) is the origin. The line \(l _ { 1 }\) passes through \(B\) and \(C\).

1. Find a vector equation for \(l _ { 1 }\).
   The line \(l _ { 2 }\) has equation \(\mathbf { r } = - 2 \mathbf { i } + \mathbf { j } + 4 \mathbf { k } + \mu ( 3 \mathbf { i } + \mathbf { j } - 2 \mathbf { k } )\).
2. Find the coordinates of the point of intersection of \(l _ { 1 }\) and \(l _ { 2 }\).
3. The point \(D\) on \(l _ { 2 }\) is such that \(\mathrm { AB } = \mathrm { BD }\). Find the position vector of \(D\). \includegraphics[max width=\textwidth, alt={}, center]{5eb2657c-ed74-4ed2-b8c4-08e9e0f657b5-13_58_1545_388_349}

10 The equations of two straight lines are $$\mathbf { r } = \mathbf { i } + \mathbf { j } + 2 a \mathbf { k } + \lambda ( 3 \mathbf { i } + 4 \mathbf { j } + a \mathbf { k } ) \quad \text { and } \quad \mathbf { r } = - 3 \mathbf { i } - \mathbf { j } + 4 \mathbf { k } + \mu ( - \mathbf { i } + 2 \mathbf { j } + 2 \mathbf { k } ) ,$$ where \(a\) is a constant.

1. Given that the acute angle between the directions of these lines is \(\frac { 1 } { 4 } \pi\), find the possible values of \(a\). \includegraphics[max width=\textwidth, alt={}, center]{b1c4d339-322f-496d-833e-8b2d002d7c48-14_2715_35_144_2012}
2. Given instead that the lines intersect, find the value of \(a\) and the position vector of the point of intersection.

10 The points \(A\) and \(B\) have position vectors \(2 \mathbf { i } + \mathbf { j } + \mathbf { k }\) and \(\mathbf { i } - 2 \mathbf { j } + 2 \mathbf { k }\) respectively. The line \(l\) has vector equation \(\mathbf { r } = \mathbf { i } + 2 \mathbf { j } - 3 \mathbf { k } + \mu ( \mathbf { i } - 3 \mathbf { j } - 2 \mathbf { k } )\).

1. Find a vector equation for the line through \(A\) and \(B\).
2. Find the acute angle between the directions of \(A B\) and \(l\), giving your answer in degrees.
3. Show that the line through \(A\) and \(B\) does not intersect the line \(l\).

11 Two lines have equations \(\mathbf { r } = \mathbf { i } + 2 \mathbf { j } + \mathbf { k } + \lambda ( a \mathbf { i } + 2 \mathbf { j } - \mathbf { k } )\) and \(\mathbf { r } = 2 \mathbf { i } + \mathbf { j } - \mathbf { k } + \mu ( 2 \mathbf { i } - \mathbf { j } + \mathbf { k } )\), where \(a\) is a constant.

1. Given that the two lines intersect, find the value of \(a\) and the position vector of the point of intersection.
2. Given instead that the acute angle between the directions of the two lines is \(\cos ^ { - 1 } \left( \frac { 1 } { 6 } \right)\), find the two possible values of \(a\).
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

9 With respect to the origin \(O\), the position vectors of the points \(A , B\) and \(C\) are given by $$\overrightarrow { O A } = \left( \begin{array} { l } 0 \\ 5 \\ 2 \end{array} \right) , \quad \overrightarrow { O B } = \left( \begin{array} { l } 1 \\ 0 \\ 1 \end{array} \right) \quad \text { and } \quad \overrightarrow { O C } = \left( \begin{array} { r } 4 \\ - 3 \\ - 2 \end{array} \right)$$ The midpoint of \(A C\) is \(M\) and the point \(N\) lies on \(B C\), between \(B\) and \(C\), and is such that \(B N = 2 N C\).

1. Find the position vectors of \(M\) and \(N\).
2. Find a vector equation for the line through \(M\) and \(N\).
3. Find the position vector of the point \(Q\) where the line through \(M\) and \(N\) intersects the line through \(A\) and \(B\).

11 The line \(l\) has equation \(\mathbf { r } = \mathbf { i } - 2 \mathbf { j } - 3 \mathbf { k } + \lambda ( - \mathbf { i } + \mathbf { j } + 2 \mathbf { k } )\). The points \(A\) and \(B\) have position vectors \(- 2 \mathbf { i } + 2 \mathbf { j } - \mathbf { k }\) and \(3 \mathbf { i } - \mathbf { j } + \mathbf { k }\) respectively.

1. Find a unit vector in the direction of \(l\).
   The line \(m\) passes through the points \(A\) and \(B\).
2. Find a vector equation for \(m\).
3. Determine whether lines \(l\) and \(m\) are parallel, intersect or are skew.
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

8. With respect to a fixed origin \(O\), the lines \(l _ { 1 }\) and \(l _ { 2 }\) are given by the equations $$l _ { 1 } : \mathbf { r } = \left( \begin{array} { r } - 1 \\ 5 \\ 4 \end{array} \right) + \lambda \left( \begin{array} { r } 2 \\ - 1 \\ 5 \end{array} \right) \quad l _ { 2 } : \mathbf { r } = \left( \begin{array} { r } 2 \\ - 2 \\ - 5 \end{array} \right) + \mu \left( \begin{array} { r } 4 \\ - 3 \\ b \end{array} \right)$$ where \(\lambda\) and \(\mu\) are scalar parameters and \(b\) is a constant.
Prove that for all values of \(b \neq 7\), the lines \(l _ { 1 }\) and \(l _ { 2 }\) are skew.

5. 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 } 4
4
- 5 \end{array} \right) + \lambda \left( \begin{array} { r } 2
- 3

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

$$\begin{aligned} & l _ { 1 } : \mathbf { r } = \left( \begin{array} { r } 2
8

10 \end{array} \right) + \lambda \left( \begin{array} { r } - 1
2
3 \end{array} \right)
& l _ { 2 } : \mathbf { r } = \left( \begin{array} { r } - 4
- 1
2 \end{array} \right) + \mu \left( \begin{array} { l } 5
4
8 \end{array} \right) \end{aligned}$$ where \(\lambda\) and \(\mu\) are scalar parameters.
Prove that lines \(l _ { 1 }\) and \(l _ { 2 }\) are skew.

1. A spherical ball of ice of radius 12 cm is placed in a bucket of water.

In a model of the situation,

• the ball remains spherical as it melts
• \(t\) minutes after the ball of ice is placed in the bucket, its radius is \(r \mathrm {~cm}\)
• the rate of decrease of the radius of the ball of ice is inversely proportional to the square of the radius
• the radius of the ball of ice is 6 cm after 15 minutes

Using the model and the information given,

1. find an equation linking \(r\) and \(t\),
2. find the time taken for the ball of ice to melt completely.
3. On Diagram 1 on page 27, sketch a graph of \(r\) against \(t\). \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{032d2541-9905-4570-9584-9a144b02fde5-27_662_728_1959_671} \captionsetup{labelformat=empty} \caption{Diagram 1}
   \end{figure}

9 The lines \(l\) and \(m\) have equations \(\mathbf { r } = 3 \mathbf { i } - 2 \mathbf { j } + \mathbf { k } + \lambda ( - \mathbf { i } + 2 \mathbf { j } + \mathbf { k } )\) and \(\mathbf { r } = 4 \mathbf { i } + 4 \mathbf { j } + 2 \mathbf { k } + \mu ( a \mathbf { i } + b \mathbf { j } - \mathbf { k } )\) respectively, where \(a\) and \(b\) are constants.

1. Given that \(l\) and \(m\) intersect, show that $$2 a - b = 4 .$$
2. Given also that \(l\) and \(m\) are perpendicular, find the values of \(a\) and \(b\).
3. When \(a\) and \(b\) have these values, find the position vector of the point of intersection of \(l\) and \(m\).

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 \(\frac { 1 } { 3 } \sqrt { 5 }\).

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 \(l _ { 2 }\).

7. Relative to a fixed origin, two lines have the equations
and $$\begin{aligned} & \mathbf { r } = \left( \begin{array} { c } 4 \\ 1 \\ 1 \end{array} \right) + s \left( \begin{array} { l } 1 \\ 4 \\ 5 \end{array} \right) \\ & \mathbf { r } = \left( \begin{array} { c } - 3 \\ 1 \\ - 6 \end{array} \right) + t \left( \begin{array} { l } 3 \\ a \\ b \end{array} \right) , \end{aligned}$$ where \(a\) and \(b\) are constants and \(s\) and \(t\) are scalar parameters.
Given that the two lines are perpendicular,

1. find a linear relationship between \(a\) and \(b\). Given also that the two lines intersect,
2. find the values of \(a\) and \(b\),
3. find the coordinates of the point where they intersect.

5．The point \(A\) has position vector \(7 \mathbf { i } + 2 \mathbf { j } - 7 \mathbf { k }\) and the point \(B\) has position vector \(12 \mathbf { i } + 3 \mathbf { j } - 15 \mathbf { k }\) ．
（a）Find a vector for the line \(L _ { 1 }\) which passes through \(A\) and \(B\) ． The line \(L _ { 2 }\) has vector equation $$\mathbf { r } = - 4 \mathbf { i } + 12 \mathbf { k } + \mu ( \mathbf { i } - 3 \mathbf { k } )$$ （b）Show that \(L _ { 2 }\) passes through the origin \(O\) ．
（c）Show that \(L _ { 1 }\) and \(L _ { 2 }\) intersect at a point \(C\) and find the position vector of \(C\) ．
（d）Find the cosine of \(\angle O C A\) ．
（e）Hence，or otherwise，find the shortest distance from \(O\) to \(L _ { 1 }\) ．
（f）Show that \(| \overrightarrow { C O } | = | \overrightarrow { A B } |\) ．
（g）Find a vector equation for the line which bisects \(\angle O C A\) ． \includegraphics[max width=\textwidth, alt={}, center]{f9d3e02c-cef2-435b-9cda-76c43fcac575-4_922_1054_279_586} Figure 1 shows a sketch of part of the curve with equation \(y = \mathrm { f } ( x )\), where $$\mathrm { f } ( x ) = x \left( 12 - x ^ { 2 } \right) .$$ The curve cuts the \(x\)-axis at the points \(P , O\) and \(R\), and \(Q\) is the maximum point.

4. \begin{figure}[h] \end{figure} Figure 1 shows a cuboid \(O A B C D E F G\), where \(O\) is the origin, \(A\) has position vector \(5 \mathbf { i } , C\) has position vector \(10 \mathbf { j }\) and \(D\) has position vector \(20 \mathbf { k }\).

1. Find the cosine of angle \(C A F\). Given that the point \(X\) lies on \(A C\) and that \(F X\) is perpendicular to \(A C\),
2. find the position vector of point \(X\) and the distance \(F X\). The line \(l _ { 1 }\) passes through \(O\) and through the midpoint of the face \(A B F E\). The line \(l _ { 2 }\) passes through \(A\) and through the midpoint of the edge \(F G\).
3. Show that \(l _ { 1 }\) and \(l _ { 2 }\) intersect and find the coordinates of the point of intersection.

3．The lines \(L _ { 1 }\) and \(L _ { 2 }\) have equations given by \(L _ { 1 } : \quad \mathbf { r } = \left( \begin{array} { c } - 7 \\ 7 \\ 1 \end{array} \right) + \lambda \left( \begin{array} { c } 2 \\ 0 \\ - 3 \end{array} \right)\) and \(L _ { 2 } : \quad \mathbf { r } = \left( \begin{array} { c } 7 \\ p \\ - 6 \end{array} \right) + \mu \left( \begin{array} { c } 10 \\ - 4 \\ - 1 \end{array} \right)\) where \(\lambda\) and \(\mu\) are parameters and \(p\) is a constant．
The two lines intersect at the point \(C\) ．
（a）Find
（i）the value of \(p\) ，
（ii）the position vector of \(C\) ．
（b）Show that the point \(B\) with position vector \(\left( \begin{array} { c } - 13 \\ 11 \\ - 4 \end{array} \right)\) lies on \(L _ { 2 }\) ． The point \(A\) with position vector \(\left( \begin{array} { c } - 7 \\ 7 \\ 1 \end{array} \right)\) lies on \(L _ { 1 }\) ．
（c）Find \(\cos ( \angle A C B )\) ，giving your answer as an exact fraction． The line \(L _ { 3 }\) bisects the angle \(A C B\) ．
（d）Find a vector equation of \(L _ { 3 }\) ．