4.04e Line intersections: parallel, skew, or intersecting

10 Lines \(L _ { 1 } , L _ { 2 }\) and \(L _ { 3 }\) have vector equations $$\begin{aligned} & L _ { 1 } = ( 4 \mathbf { i } + \mathbf { j } + 3 \mathbf { k } ) + s ( 6 \mathbf { i } + 9 \mathbf { j } - 3 \mathbf { k } ) , \\ & L _ { 2 } = ( 2 \mathbf { i } + 3 \mathbf { j } ) + t ( - 3 \mathbf { i } - 8 \mathbf { j } + 6 \mathbf { k } ) , \\ & L _ { 3 } = ( 3 \mathbf { i } - \mathbf { j } + 4 \mathbf { k } ) + u ( - 2 \mathbf { i } + c \mathbf { j } + \mathbf { k } ) . \end{aligned}$$ In each of the following cases, find the value of \(c\).

1. \(\quad L _ { 1 }\) and \(L _ { 3 }\) are parallel.
2. \(\quad L _ { 2 }\) and \(L _ { 3 }\) intersect.

9 Determine whether the lines whose equations are $$\mathbf { r } = ( 4 + 2 \mu ) \mathbf { i } + ( 7 + 3 \mu ) \mathbf { j } + ( 3 + 7 \mu ) \mathbf { k } \quad \text { and } \quad \mathbf { r } = ( 35 - 5 \lambda ) \mathbf { i } + ( 6 + 2 \lambda ) \mathbf { j } + ( 14 + 3 \lambda ) \mathbf { k }$$ intersect, are parallel or are skew.

4 A sequence of complex numbers is defined by $$u _ { 1 } = 1 + \mathrm { i } \quad \text { and } \quad u _ { n + 1 } = \mathrm { i } u _ { n } ( n = 1,2,3 , \ldots )$$

1. Find \(u _ { 2 } , u _ { 3 } , u _ { 4 } , u _ { 5 }\) and \(u _ { 6 }\).
2. Describe the behaviour of the sequence.
3. Hence evaluate \(\sum _ { n = 1 } ^ { 73 } u _ { n }\).

6 Two straight lines have equations $$\mathbf { r } = - 3 \mathbf { i } + 11 \mathbf { j } - 9 \mathbf { k } + \lambda ( 4 \mathbf { i } + 7 \mathbf { j } + 8 \mathbf { k } )$$ and $$\mathbf { r } = 21 \mathbf { i } + 2 \mathbf { j } + 15 \mathbf { k } + \mu ( 3 \mathbf { i } + \mathbf { j } + 4 \mathbf { k } )$$

1. Show that the lines intersect and find the coordinates of their point of intersection.
2. Find the acute angle between the two lines.

8

1. Show that the lines $$\mathbf { r } = - 3 \mathbf { i } + \mathbf { j } - 5 \mathbf { k } + \lambda ( \mathbf { i } + 2 \mathbf { j } + \mathbf { 6 } \mathbf { k } ) \quad \text { and } \quad \mathbf { r } = 4 \mathbf { i } + \mathbf { j } + 5 \mathbf { k } + \mu ( - 3 \mathbf { i } + \mathbf { j } - 2 \mathbf { k } )$$ intersect and write down the coordinates of their point of intersection.
2. Find in degrees the obtuse angle between the two lines.

10 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 equations of two straight lines are $$\mathbf{r} = \mathbf{i} + \mathbf{j} + 2a\mathbf{k} + \lambda(3\mathbf{i} + 4\mathbf{j} + a\mathbf{k}) \quad \text{and} \quad \mathbf{r} = -3\mathbf{i} - \mathbf{j} + 4\mathbf{k} + \mu(-\mathbf{i} + 2\mathbf{j} + 2\mathbf{k}),$$ 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 \(a\). [6]
2. Given instead that the lines intersect, find the value of \(a\) and the position vector of the point of intersection. [5]

Two lines have equations \(\mathbf{r} = \begin{pmatrix} 1 \\ 3 \\ 2 \end{pmatrix} + s \begin{pmatrix} 2 \\ -1 \\ 3 \end{pmatrix}\) and \(\mathbf{r} = \begin{pmatrix} 2 \\ 1 \\ 4 \end{pmatrix} + t \begin{pmatrix} 1 \\ -1 \\ 4 \end{pmatrix}\).

1. Show that the lines are skew. [5]
2. Find the acute angle between the directions of the two lines. [3]

The equations of two lines \(l\) and \(m\) are \(\mathbf{r} = 3\mathbf{i} - \mathbf{j} - 2\mathbf{k} + \lambda(-\mathbf{i} + \mathbf{j} + 4\mathbf{k})\) and \(\mathbf{r} = 4\mathbf{i} + 4\mathbf{j} - 3\mathbf{k} + \mu(2\mathbf{i} + \mathbf{j} - 2\mathbf{k})\) respectively.

1. Show that the lines do not intersect. [3]
2. Calculate the acute angle between the directions of the lines. [3]
3. Find the equation of the plane which passes through the point \((3, -2, -1)\) and which is parallel to both \(l\) and \(m\). Give your answer in the form \(ax + by + cz = d\). [5]

The line \(l\) has equation \(\mathbf{r} = 5\mathbf{i} - 3\mathbf{j} - \mathbf{k} + \lambda(\mathbf{i} - 2\mathbf{j} + \mathbf{k})\). The plane \(p\) has equation $$(\mathbf{r} - \mathbf{i} - 2\mathbf{j}) \cdot (3\mathbf{i} + \mathbf{j} + \mathbf{k}) = 0.$$ The line \(l\) intersects the plane \(p\) at the point \(A\).

1. Find the position vector of \(A\). [3]
2. Calculate the acute angle between \(l\) and \(p\). [4]
3. Find the equation of the line which lies in \(p\) and intersects \(l\) at right angles. [4]

With respect to a fixed origin \(O\), the equations of lines \(l_1\) and \(l_2\) are given by $$l_1: \mathbf{r} = \begin{pmatrix} 2 \\ 8 \\ 10 \end{pmatrix} + \lambda \begin{pmatrix} -1 \\ 2 \\ 3 \end{pmatrix}$$ $$l_2: \mathbf{r} = \begin{pmatrix} -4 \\ -1 \\ 2 \end{pmatrix} + \mu \begin{pmatrix} 5 \\ 4 \\ 8 \end{pmatrix}$$ where \(\lambda\) and \(\mu\) are scalar parameters. Prove that lines \(l_1\) and \(l_2\) are skew. [5]

The line \(l_1\) has vector equation $$\mathbf{r} = \begin{pmatrix} 3 \\ 1 \\ 2 \end{pmatrix} + \lambda \begin{pmatrix} 1 \\ -1 \\ 4 \end{pmatrix}$$ and the line \(l_2\) has vector equation $$\mathbf{r} = \begin{pmatrix} 0 \\ 4 \\ -2 \end{pmatrix} + \mu \begin{pmatrix} 1 \\ -1 \\ 0 \end{pmatrix},$$ where \(\lambda\) and \(\mu\) are parameters. The lines \(l_1\) and \(l_2\) intersect at the point \(B\) and the acute angle between \(l_1\) and \(l_2\) is \(\theta\).

1. Find the coordinates of \(B\). [4]
2. Find the value of \(\cos \theta\), giving your answer as a simplified fraction. [4]

The point \(A\), which lies on \(l_1\), has position vector \(\mathbf{a} = 3\mathbf{i} + \mathbf{j} + 2\mathbf{k}\). The point \(C\), which lies on \(l_2\), has position vector \(\mathbf{c} = 5\mathbf{i} - \mathbf{j} - 2\mathbf{k}\). The point \(D\) is such that \(ABCD\) is a parallelogram.

1. Show that \(|\overrightarrow{AB}| = |\overrightarrow{BC}|\). [3]
2. Find the position vector of the point \(D\). [2]

Two submarines are travelling in straight lines through the ocean. Relative to a fixed origin, the vector equations of the two lines, \(l_1\) and \(l_2\), along which they travel are $$\mathbf{r} = 3\mathbf{i} + 4\mathbf{j} - 5\mathbf{k} + \lambda(\mathbf{i} - 2\mathbf{j} + 2\mathbf{k})$$ and \(\mathbf{r} = 9\mathbf{i} + \mathbf{j} - 2\mathbf{k} + \mu (4\mathbf{i} + \mathbf{j} - \mathbf{k})\), where \(\lambda\) and \(\mu\) are scalars.

1. Show that the submarines are moving in perpendicular directions. [2]
2. Given that \(l_1\) and \(l_2\) intersect at the point \(A\), find the position vector of \(A\). [5]

The point \(B\) has position vector \(10\mathbf{j} - 11\mathbf{k}\).

1. Show that only one of the submarines passes through the point \(B\). [3]
2. Given that 1 unit on each coordinate axis represents 100 m, find, in km, the distance \(AB\). [2]

The line \(l_1\) passes through the point \(A(0, 6, 9)\) and the point \(B(4, -6, -11)\). The line \(l_2\) has equation \(\mathbf{r} = \begin{bmatrix} -1 \\ 5 \\ -2 \end{bmatrix} + \lambda \begin{bmatrix} 3 \\ -5 \\ 1 \end{bmatrix}\).

1. The acute angle between the lines \(l_1\) and \(l_2\) is \(\theta\). Find the value of \(\cos \theta\) as a fraction in its lowest terms. [5 marks]
2. Show that the lines \(l_1\) and \(l_2\) intersect and find the coordinates of the point of intersection. [5 marks]
3. The points \(C\) and \(D\) lie on line \(l_2\) such that \(ACBD\) is a parallelogram. \includegraphics{figure_6} The length of \(AB\) is three times the length of \(CD\). Find the coordinates of the points \(C\) and \(D\). [5 marks]

Two submarines are travelling in straight lines through the ocean. Relative to a fixed origin, the vector equations of the two lines, \(l_1\) and \(l_2\), along which they travel are \begin{align} \mathbf{r} &= 3\mathbf{i} + 4\mathbf{j} - 5\mathbf{k} + \lambda(\mathbf{i} - 2\mathbf{j} + 2\mathbf{k})
\text{and} \quad \mathbf{r} &= 9\mathbf{i} + \mathbf{j} - 2\mathbf{k} + \mu (4\mathbf{i} + \mathbf{j} - \mathbf{k}), \end{align} where \(\lambda\) and \(\mu\) are scalars.

1. Show that the submarines are moving in perpendicular directions. [2]
2. Given that \(l_1\) and \(l_2\) intersect at the point \(A\), find the position vector of \(A\). [5]

The point \(B\) has position vector \(10\mathbf{j} - 11\mathbf{k}\).

1. Show that only one of the submarines passes through the point \(B\). [3]
2. Given that 1 unit on each coordinate axis represents 100 m, find, in km, the distance \(AB\). [2]

The line \(L_1\) passes through the points \((2, -3, 1)\) and \((-1, -2, -4)\). The line \(L_2\) passes through the point \((3, 2, -9)\) and is parallel to the vector \(\mathbf{4i} - \mathbf{4j} + \mathbf{5k}\).

1. Find an equation for \(L_1\) in the form \(\mathbf{r} = \mathbf{a} + t\mathbf{b}\). [2]
2. Prove that \(L_1\) and \(L_2\) are skew. [5]

Two lines have vector equations $$\mathbf{r} = \mathbf{i} - 2\mathbf{j} + 4\mathbf{k} + \lambda(3\mathbf{i} + \mathbf{j} + a\mathbf{k})$$ and $$\mathbf{r} = -8\mathbf{i} + 2\mathbf{j} + 3\mathbf{k} + \mu(\mathbf{i} - 2\mathbf{j} - \mathbf{k}),$$ where \(a\) is a constant.

1. Given that the lines are skew, find the value that \(a\) cannot take. [6]
2. Given instead that the lines intersect, find the point of intersection. [2]

Show that the straight lines with equations \(\mathbf{r} = \begin{pmatrix} 4 \\ 2 \\ 4 \end{pmatrix} + \lambda \begin{pmatrix} 3 \\ 0 \\ 1 \end{pmatrix}\) and \(\mathbf{r} = \begin{pmatrix} -1 \\ 4 \\ 9 \end{pmatrix} + \mu \begin{pmatrix} -1 \\ 1 \\ 3 \end{pmatrix}\) meet. Find their point of intersection. [5]

Relative to a fixed origin, two lines have the equations $$\mathbf{r} = \begin{pmatrix} 4 \\ 1 \\ 1 \end{pmatrix} + s \begin{pmatrix} 1 \\ 4 \\ 5 \end{pmatrix}$$ and $$\mathbf{r} = \begin{pmatrix} -3 \\ 1 \\ -6 \end{pmatrix} + t \begin{pmatrix} 3 \\ a \\ b \end{pmatrix},$$ 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 \(b\). [2]

Given also that the two lines intersect,

1. find the values of \(a\) and \(b\), [8]
2. find the coordinates of the point where they intersect. [2]

Relative to a fixed origin, two lines have the equations $$\mathbf{r} = (7\mathbf{i} - 4\mathbf{k}) + s(4\mathbf{i} - 3\mathbf{j} + \mathbf{k}),$$ and $$\mathbf{r} = (-7\mathbf{i} + \mathbf{j} + 8\mathbf{k}) + t(-3\mathbf{i} + 2\mathbf{k}),$$ where \(s\) and \(t\) are scalar parameters.

1. Show that the two lines intersect and find the position vector of the point where they meet. [5]
2. Find, in degrees to 1 decimal place, the acute angle between the lines. [4]

Relative to a fixed origin, the points \(A\) and \(B\) have position vectors \(\begin{pmatrix} -4 \\ 1 \\ 3 \end{pmatrix}\) and \(\begin{pmatrix} -3 \\ 6 \\ 1 \end{pmatrix}\) respectively.

1. Find a vector equation for the line \(l_1\) which passes through \(A\) and \(B\). [2]

The line \(l_2\) has vector equation $$\mathbf{r} = \begin{pmatrix} 3 \\ -7 \\ 9 \end{pmatrix} + \mu \begin{pmatrix} 2 \\ -3 \\ 1 \end{pmatrix}.$$

1. Show that lines \(l_1\) and \(l_2\) do not intersect. [5]
2. Find the position vector of the point \(C\) on \(l_2\) such that \(\angle ABC = 90°\). [6]

The line \(l_1\) passes through the points \(A\) and \(B\) with position vectors \((\mathbf{3i} + \mathbf{6j} - \mathbf{8k})\) and \((\mathbf{8j} - \mathbf{6k})\) respectively, relative to a fixed origin.

1. Find a vector equation for \(l_1\). [2]

The line \(l_2\) has vector equation $$\mathbf{r} = (-\mathbf{2i} + \mathbf{10j} + \mathbf{6k}) + \mu(\mathbf{7i} - \mathbf{4j} + \mathbf{6k}),$$ where \(\mu\) is a scalar parameter.

1. Show that lines \(l_1\) and \(l_2\) intersect. [4]
2. Find the coordinates of the point where \(l_1\) and \(l_2\) intersect. [2]

The point \(C\) lies on \(l_2\) and is such that \(AC\) is perpendicular to \(AB\).

1. Find the position vector of \(C\). [6]

Relative to a fixed origin, two lines have the equations $$\mathbf{r} = (7\mathbf{i} - 4\mathbf{k}) + s(4\mathbf{i} - 3\mathbf{j} + \mathbf{k}),$$ and $$\mathbf{r} = (-7\mathbf{i} + \mathbf{j} + 8\mathbf{k}) + t(-3\mathbf{i} + 2\mathbf{k}),$$ where \(s\) and \(t\) are scalar parameters.

1. Show that the two lines intersect and find the position vector of the point where they meet. [5]
2. Find, in degrees to 1 decimal place, the acute angle between the lines. [4]

A straight road passes through villages at the points \(A\) and \(B\) with position vectors \((9\mathbf{i} - 8\mathbf{j} + 2\mathbf{k})\) and \((4\mathbf{j} + \mathbf{k})\) respectively, relative to a fixed origin. The road ends at a junction at the point \(C\) with another straight road which lies along the line with equation $$\mathbf{r} = (2\mathbf{i} + 16\mathbf{j} - \mathbf{k}) + t(-5\mathbf{i} + 3\mathbf{j}),$$ where \(t\) is a scalar parameter.

1. Find the position vector of \(C\). [5]

Given that 1 unit on each coordinate axis represents 200 metres,

1. find the distance, in kilometres, from the village at \(A\) to the junction at \(C\). [4]

A line \(l\) has equation \(\mathbf{r} = 3\mathbf{i} + \mathbf{j} - 2\mathbf{k} + t(\mathbf{i} + 4\mathbf{j} + 2\mathbf{k})\) and a plane \(\Pi\) has equation \(8x - 7y + 10z = 7\). Determine whether \(l\) lies in \(\Pi\), is parallel to \(\Pi\) without intersecting it, or intersects \(\Pi\) at one point. [5]