4.04e Line intersections: parallel, skew, or intersecting

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 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.

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 \(C\) .

1. Find
   1. the value of \(p\) ,
   2. the position vector of \(C\) .
2. 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 \(L _ { 1 }\) .
3. Find \(\cos ( \angle A C B )\) ,giving your answer as an exact fraction. The line \(L _ { 3 }\) bisects the angle \(A C B\) .
4. Find a vector equation of \(L _ { 3 }\) .

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.

3 A line \(l\) has equation \(\frac { x - 6 } { - 4 } = \frac { y + 7 } { 8 } = \frac { z + 10 } { 7 }\) and a plane \(p\) has equation \(3 x - 4 y - 2 z = 8\).

1. Find the point of intersection of \(l\) and \(p\).
2. Find the equation of the plane which contains \(l\) and is perpendicular to \(p\), giving your answer in the form \(a x + b y + c z = d\).

6 The line \(l _ { 1 }\) has equation \(\mathbf { r } = \left( \begin{array} { r } 3 \\ 0 \\ - 2 \end{array} \right) + s \left( \begin{array} { r } 2 \\ 3 \\ - 4 \end{array} \right)\). The line \(l _ { 2 }\) has equation \(\mathbf { r } = \left( \begin{array} { l } 5 \\ 3 \\ 2 \end{array} \right) + t \left( \begin{array} { r } 0 \\ 1 \\ - 2 \end{array} \right)\).

1. Find the acute angle between \(l _ { 1 }\) and \(l _ { 2 }\).
2. Show that \(l _ { 1 }\) and \(l _ { 2 }\) are skew.
3. One of the numbers in the equation of line \(l _ { 1 }\) is changed so that the equation becomes \(\mathbf { r } = \left( \begin{array} { l } 3 \\ 0 \\ a \end{array} \right) + s \left( \begin{array} { r } 2 \\ 3 \\ - 4 \end{array} \right)\). Given that \(l _ { 1 }\) and \(l _ { 2 }\) now intersect, find \(a\).

4 The equations of two lines are $$\mathbf { r } = \mathbf { i } + 2 \mathbf { j } + \lambda ( 2 \mathbf { i } + \mathbf { j } + 3 \mathbf { k } ) \text { and } \mathbf { r } = 6 \mathbf { i } + 8 \mathbf { j } + \mathbf { k } + \mu ( \mathbf { i } + 4 \mathbf { j } - 5 \mathbf { k } ) .$$

1. Show that these lines meet, and find the coordinates of the point of intersection.
2. Find the acute angle between these lines.

5 The equations of three lines are as follows. $$\begin{array} { l l } \text { Line } A : & \mathbf { r } = \mathbf { i } + 4 \mathbf { j } + \mathbf { k } + s ( - \mathbf { i } + 2 \mathbf { j } + 2 \mathbf { k } ) \\ \text { Line } B : & \mathbf { r } = 2 \mathbf { i } + 8 \mathbf { j } + 2 \mathbf { k } + t ( \mathbf { i } + 3 \mathbf { j } + 5 \mathbf { k } ) \\ \text { Line } C : & \mathbf { r } = - \mathbf { i } + 19 \mathbf { j } + 15 \mathbf { k } + u ( 2 \mathbf { i } - 4 \mathbf { j } - 4 \mathbf { k } ) \end{array}$$

1. Show that lines \(A\) and \(B\) are skew.
2. Determine, giving reasons, the geometrical relationship between lines \(A\) and \(C\).

5 The vector equations of two lines are as follows. $$L : \mathbf { r } = \left( \begin{array} { l } 1 \\ 4 \\ 5 \end{array} \right) + s \left( \begin{array} { c } 2 \\ - 1 \\ 3 \end{array} \right) \quad M : \mathbf { r } = \left( \begin{array} { c } 3 \\ 2 \\ - 5 \end{array} \right) + t \left( \begin{array} { c } 5 \\ - 3 \\ 1 \end{array} \right)$$

1. Show that the lines \(L\) and \(M\) meet, and find the coordinates of the point of intersection.
2. Show that the line \(L\) can also be represented by the equation \(\mathbf { r } = \left( \begin{array} { c } 7 \\ 1 \\ 14 \end{array} \right) + u \left( \begin{array} { c } - 4 \\ 2 \\ - 6 \end{array} \right)\).

5

1. Verify that the lines \(\mathbf { r } = \left( \begin{array} { r } - 5 \\ 3 \\ 4 \end{array} \right) + \lambda \left( \begin{array} { r } 3 \\ 0 \\ - 1 \end{array} \right)\) and \(\mathbf { r } = \left( \begin{array} { r } - 1 \\ 4 \\ 2 \end{array} \right) + \mu \left( \begin{array} { r } 2 \\ - 1 \\ 0 \end{array} \right)\) meet at the point (1,3,2).
2. Find the acute angle between the lines.

6 Fig. 6 shows a lean-to greenhouse ABCDHEFG . With respect to coordinate axes Oxyz , the coordinates of the vertices are as shown. All distances are in metres. Ground level is the plane \(z = 0\). \begin{figure}[h] \end{figure}

1. Verify that the equation of the plane through \(\mathrm { A } , \mathrm { B }\) and E is \(x + 6 y + 12 = 0\). Hence, given that F lies in this plane, show that \(a = - 2 \frac { 1 } { 3 }\).
2. (A) Show that the vector \(\left( \begin{array} { r } 1 \\ - 6 \\ 0 \end{array} \right)\) is normal to the plane DHC.
   (B) Hence find the cartesian equation of this plane.
   (C) Given that G lies in the plane DHC , find \(b\) and the length FG .
3. Find the angle EFB . A straight wire joins point H to a point P which is half way between E and F . Q is a point two-thirds of the way along this wire, so that \(\mathrm { HQ } = 2 \mathrm { QP }\).
4. Find the height of Q above the ground. \section*{Question 7 begins on page 4.}

10 The sequence \(u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots\) is defined by \(u _ { n } = 5 ^ { n } + 2 ^ { n - 1 }\).

1. Find \(u _ { 1 } , u _ { 2 }\) and \(u _ { 3 }\).
2. Hence suggest a positive integer, other than 1 , which divides exactly into every term of the sequence.
3. By considering \(u _ { n + 1 } + u _ { n }\), prove by induction that your suggestion in part (ii) is correct. \section*{OCR}

7 The lines \(l _ { 1 }\) and \(l _ { 2 }\) have vector equations $$\mathbf { r } = 4 \mathbf { i } - 2 \mathbf { j } + \lambda ( 2 \mathbf { i } + \mathbf { j } - 4 \mathbf { k } ) \quad \text { and } \quad \mathbf { r } = 4 \mathbf { i } - 5 \mathbf { j } + 2 \mathbf { k } + \mu ( \mathbf { i } - \mathbf { j } - \mathbf { k } )$$ respectively.

1. Show that \(l _ { 1 }\) and \(l _ { 2 }\) intersect.
2. Find the perpendicular distance from the point \(P\) whose position vector is \(3 \mathbf { i } - 5 \mathbf { j } + 6 \mathbf { k }\) to the plane containing \(l _ { 1 }\) and \(l _ { 2 }\).
3. Find the perpendicular distance from \(P\) to \(l _ { 1 }\).

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.
   }

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 \(a\).

7 The lines \(l _ { 1 }\) and \(l _ { 2 }\) have vector equations $$\mathbf { r } = a \mathbf { i } + 9 \mathbf { j } + 13 \mathbf { k } + \lambda ( \mathbf { i } + 2 \mathbf { j } + 3 \mathbf { k } ) \quad \text { and } \quad \mathbf { r } = - 3 \mathbf { i } + 7 \mathbf { j } - 2 \mathbf { k } + \mu ( - \mathbf { i } + 2 \mathbf { j } - 3 \mathbf { k } )$$ respectively. It is given that \(l _ { 1 }\) and \(l _ { 2 }\) intersect.

1. Find the value of the constant \(a\).
   The point \(P\) has position vector \(3 \mathbf { i } + \mathbf { j } + 6 \mathbf { k }\).
2. Find the perpendicular distance from \(P\) to the plane containing \(l _ { 1 }\) and \(l _ { 2 }\).
3. Find the perpendicular distance from \(P\) to \(l _ { 2 }\).

8 The plane \(\Pi _ { 1 }\) has equation \(\mathbf { r } = \left( \begin{array} { r } 2 \\ 3 \\ - 1 \end{array} \right) + s \left( \begin{array} { l } 1 \\ 0 \\ 1 \end{array} \right) + t \left( \begin{array} { r } 1 \\ - 1 \\ - 2 \end{array} \right)\). Find a cartesian equation of \(\Pi _ { 1 }\). The plane \(\Pi _ { 2 }\) has equation \(2 x - y + z = 10\). Find the acute angle between \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\). Find an equation of the line of intersection of \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\), giving your answer in the form \(\mathbf { r } = \mathbf { a } + \lambda \mathbf { b }\).

16
10
22 \end{array} \right)$$ has the form $$\mathbf { x } = \left( \begin{array} { r } 1
- 2
- 3
- 4 \end{array} \right) + \lambda \mathbf { e } _ { 1 } + \mu \mathbf { e } _ { 2 } ,$$ where \(\lambda\) and \(\mu\) are real numbers and \(\left\{ \mathbf { e } _ { 1 } , \mathbf { e } _ { 2 } \right\}\) is a basis for \(K\). 7 The square matrix \(\mathbf { A }\) has \(\lambda\) as an eigenvalue with \(\mathbf { e }\) as a corresponding eigenvector. Show that \(\mathbf { e }\) is an eigenvector of \(\mathbf { A } ^ { 2 }\) and state the corresponding eigenvalue. Find the eigenvalues of the matrix \(\mathbf { B }\), where $$\mathbf { B } = \left( \begin{array} { l l l } 1 & 3 & 0
2 & 0 & 2
1 & 1 & 2 \end{array} \right)$$ Find the eigenvalues of \(\mathbf { B } ^ { 4 } + 2 \mathbf { B } ^ { 2 } + 3 \mathbf { I }\), where \(\mathbf { I }\) is the \(3 \times 3\) identity matrix. 8 The plane \(\Pi _ { 1 }\) has equation \(\mathbf { r } = \left( \begin{array} { r } 2 \\ 3 \\ - 1 \end{array} \right) + s \left( \begin{array} { l } 1 \\ 0 \\ 1 \end{array} \right) + t \left( \begin{array} { r } 1 \\ - 1 \\ - 2 \end{array} \right)\). Find a cartesian equation of \(\Pi _ { 1 }\). The plane \(\Pi _ { 2 }\) has equation \(2 x - y + z = 10\). Find the acute angle between \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\). Find an equation of the line of intersection of \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\), giving your answer in the form \(\mathbf { r } = \mathbf { a } + \lambda \mathbf { b }\). 9 The curve \(C\) has parametric equations $$x = t ^ { 2 } , \quad y = t - \frac { 1 } { 3 } t ^ { 3 } , \quad \text { for } 0 \leqslant t \leqslant 1 .$$ Find the surface area generated when \(C\) is rotated through \(2 \pi\) radians about the \(x\)-axis. Find the coordinates of the centroid of the region bounded by \(C\), the \(x\)-axis and the line \(x = 1\). 10 The curve \(C\) has equation $$y = \frac { p x ^ { 2 } + 4 x + 1 } { x + 1 } ,$$ where \(p\) is a positive constant and \(p \neq 3\).

1. Obtain the equations of the asymptotes of \(C\).
2. Find the value of \(p\) for which the \(x\)-axis is a tangent to \(C\), and sketch \(C\) in this case.
3. For the case \(p = 1\), show that \(C\) has no turning points, and sketch \(C\), giving the exact coordinates of the points of intersection of \(C\) with the \(x\)-axis. 11 Answer only one of the following two alternatives. \section*{EITHER} State the fifth roots of unity in the form \(\cos \theta + \mathrm { i } \sin \theta\), where \(- \pi < \theta \leqslant \pi\). Simplify $$\left( x - \left[ \cos \frac { 2 } { 5 } \pi + i \sin \frac { 2 } { 5 } \pi \right] \right) \left( x - \left[ \cos \frac { 2 } { 5 } \pi - i \sin \frac { 2 } { 5 } \pi \right] \right)$$ Hence find the real factors of $$x ^ { 5 } - 1$$ Express the six roots of the equation $$x ^ { 6 } - x ^ { 3 } + 1 = 0$$ as three conjugate pairs, in the form \(\cos \theta \pm \mathrm { i } \sin \theta\). Hence find the real factors of $$x ^ { 6 } - x ^ { 3 } + 1$$ OR Given that $$y ^ { 2 } \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } - 6 y ^ { 2 } \frac { \mathrm {~d} y } { \mathrm {~d} x } + 2 y \left( \frac { \mathrm {~d} y } { \mathrm {~d} x } \right) ^ { 2 } + 3 y ^ { 3 } = 25 \mathrm { e } ^ { - 2 x }$$ and that \(v = y ^ { 3 }\), show that $$\frac { \mathrm { d } ^ { 2 } v } { \mathrm {~d} x ^ { 2 } } - 6 \frac { \mathrm {~d} v } { \mathrm {~d} x } + 9 v = 75 \mathrm { e } ^ { - 2 x }$$ Find the particular solution for \(y\) in terms of \(x\), given that when \(x = 0 , y = 2\) and \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 1\).

1 Three planes \(P , Q\) and \(R\) have the following equations. $$\begin{array} { l l } \text { Plane } P : & 8 x - y - 14 z = 20 \\ \text { Plane } Q : & 6 x + 2 y - 5 z = 26 \\ \text { Plane } R : & 2 x + y - z = 40 \end{array}$$ The line of intersection of the planes \(P\) and \(Q\) is \(K\).
The line of intersection of the planes \(P\) and \(R\) is \(L\).

1. Show that \(K\) and \(L\) are parallel lines, and find the shortest distance between them.
2. Show that the shortest distance between the line \(K\) and the plane \(R\) is \(5 \sqrt { 6 }\). The line \(M\) has equation \(\mathbf { r } = ( \mathbf { i } - 4 \mathbf { j } ) + \lambda ( 5 \mathbf { i } - 4 \mathbf { j } + 3 \mathbf { k } )\).
3. Show that the lines \(K\) and \(M\) intersect, and find the coordinates of the point of intersection.
4. Find the shortest distance between the lines \(L\) and \(M\).

1 Positions in space around an aerodrome are modelled by a coordinate system with a point on the runway as the origin, O . The \(x\)-axis is east, the \(y\)-axis is north and the \(z\)-axis is vertically upwards. Units of distance are kilometres. Units of time are hours.
At time \(t = 0\), an aeroplane, P , is at \(( 3,4,8 )\) and is travelling in a direction \(\left( \begin{array} { l } 2 \\ 1 \\ 0 \end{array} \right)\) at a constant speed of \(900 \mathrm { kmh } ^ { - 1 }\).

1. Find the least distance of the path of P from the point O . At time \(t = 0\), a second aeroplane, Q , is at \(( 80,40,10 )\). It is travelling in a straight line towards the point O . Its speed is constant at \(270 \mathrm { kmh } ^ { - 1 }\).
2. Show that the shortest distance between the paths of the two aeroplanes is 2.24 km correct to three significant figures.
3. By finding the points on the paths where the shortest distance occurs and the times at which the aeroplanes are at these points, show that in fact the aeroplanes are never this close.
4. A third aeroplane, R , is at position \(( 29,19,5.5 )\) at time \(t = 0\) and is travelling at \(285 \mathrm {~km} \mathrm {~h} ^ { - 1 }\) in a direction \(\left( \begin{array} { c } 18 \\ 6 \\ 1 \end{array} \right)\). Given that Q is in the process of landing and cannot change course, show that R needs to be instructed to alter course or change speed.

11 A 3-D coordinate system, whose units are metres, is set up to model a construction site. The construction site contains four vertical poles \(P _ { 1 } , P _ { 2 } , P _ { 3 }\) and \(P _ { 4 }\). The floor of the construction site is modelled as lying in the \(x - y\) plane and the poles are modelled as vertical line segments. One end of each pole lies on the floor of the construction site, and the other end of each pole is modelled by the points \(( 0,0,18 ) , ( 12,14,20 ) , ( 0,11,7 )\) and \(( 18,2,16 )\) respectively. A wire, \(S\), runs from the top of \(P _ { 1 }\) to the top of \(P _ { 2 }\). A second wire, \(T\), runs from the top of \(P _ { 3 }\) to the top of \(P _ { 4 }\). The wires are modelled by straight lines segments. The layout of the construction site is illustrated on the diagram below which is not drawn to scale. \includegraphics[max width=\textwidth, alt={}, center]{fbb82fa2-b316-44ae-a19e-197b45f51c87-5_707_871_696_242} A vector equation of the line segment that represents the wire \(S\) is given by \(\mathbf { r } = \left( \begin{array} { c } 0 \\ 0 \\ 18 \end{array} \right) + \lambda \left( \begin{array} { l } 6 \\ 7 \\ 1 \end{array} \right) , 0 \leqslant \lambda \leqslant 2\).

1. Find, in the same form, a vector equation of the line segment that represents the wire \(T\). The components of the direction vector should be integers whose only positive common factor is 1 . For the construction site to be considered safe, it must pass two tests.
   Test 1: The wires \(S\) and \(T\) need to be at least 5 metres apart at all positions on \(S\) and \(T\).
2. By using an appropriate formula, determine whether the construction site passes Test 1. A security camera is placed at a point \(Q\) on wire \(S\). Test 2: To ensure sufficient visibility of the construction site, the distance between the security camera and the top of \(P _ { 3 }\) must be at least 19 m .
3. Determine whether it is possible to find point \(Q\) on \(S\) such that the construction site passes Test 2.

1

1. Find a vector which is perpendicular to both \(3 \mathbf { i } - 5 \mathbf { j } - \mathbf { k }\) and \(\mathbf { i } + 3 \mathbf { j } - 4 \mathbf { k }\). The equations of two lines are \(\mathbf { r } = 2 \mathbf { i } + 3 \mathbf { j } + 3 \mathbf { k } + \lambda ( \mathbf { i } - 2 \mathbf { j } + \mathbf { k } )\) and \(\mathbf { r } = \mathbf { i } + 11 \mathbf { j } - 4 \mathbf { k } + \mu ( - \mathbf { i } + 3 \mathbf { j } - 2 \mathbf { k } )\).
2. Show that the lines intersect, stating the point of intersection.

5 The points \(A\) and \(B\) have coordinates \(( 5,1 , - 2 )\) and \(( 4 , - 1,3 )\) respectively.
The line \(l\) has equation \(\mathbf { r } = \left[ \begin{array} { r } - 8 \\ 5 \\ - 6 \end{array} \right] + \mu \left[ \begin{array} { r } 5 \\ 0 \\ - 2 \end{array} \right]\).

1. Find a vector equation of the line that passes through \(A\) and \(B\).
2. 1. Show that the line that passes through \(A\) and \(B\) intersects the line \(l\), and find the coordinates of the point of intersection, \(P\).
   2. The point \(C\) lies on \(l\) such that triangle \(P B C\) has a right angle at \(B\). Find the coordinates of \(C\).