4.04a Line equations: 2D and 3D, cartesian and vector forms

1. The plane \(\Pi _ { 1 }\) has equation

$$x - 5 y - 2 z = 3$$ The plane \(\Pi _ { 2 }\) has equation $$\mathbf { r } = \mathbf { i } + 2 \mathbf { j } + \mathbf { k } + \lambda ( \mathbf { i } + 4 \mathbf { j } + 3 \mathbf { k } ) + \mu ( 2 \mathbf { i } - \mathbf { j } + \mathbf { k } )$$ where \(\lambda\) and \(\mu\) are scalar parameters.

1. Show that \(\Pi _ { 1 }\) is perpendicular to \(\Pi _ { 2 }\)
2. Find a cartesian equation for \(\Pi _ { 2 }\)
3. Find an equation for the line of intersection of \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\) giving your answer in the form \(( \mathbf { r } - \mathbf { a } ) \times \mathbf { b } = \mathbf { 0 }\), where \(\mathbf { a }\) and \(\mathbf { b }\) are constant vectors to be found.
   (6)
   

6. The matrix \(\mathbf { M }\) is given by $$\mathbf { M } = \left( \begin{array} { r r r } 1 & k & 0 \\ 2 & - 2 & 1 \\ - 4 & 1 & - 1 \end{array} \right) , k \in \mathbb { R } , k \neq \frac { 1 } { 2 }$$

1. Show that \(\operatorname { det } \mathbf { M } = 1 - 2 k\).
2. Find \(\mathbf { M } ^ { - 1 }\) in terms of \(k\). The straight line \(l _ { 1 }\) is mapped onto the straight line \(l _ { 2 }\) by the transformation represented by the matrix $$\left( \begin{array} { r r r } 1 & 0 & 0 \\ 2 & - 2 & 1 \\ - 4 & 1 & - 1 \end{array} \right)$$ Given that \(l _ { 2 }\) has cartesian equation $$\frac { x - 1 } { 5 } = \frac { y + 2 } { 2 } = \frac { z - 3 } { 1 }$$
3. find a cartesian equation of the line \(l _ { 1 }\)

2 Find the equation of the line of intersection of the planes with equations $$\mathbf { r } . ( 3 \mathbf { i } + \mathbf { j } - 2 \mathbf { k } ) = 4 \quad \text { and } \quad \mathbf { r } . ( \mathbf { i } + 5 \mathbf { j } + 4 \mathbf { k } ) = 6 ,$$ giving your answer in the form \(\mathbf { r } = \mathbf { a } + t \mathbf { b }\).

7 The position vectors of the points \(A , B , C , D , G\) are given by $$\mathbf { a } = 6 \mathbf { i } + 4 \mathbf { j } + 8 \mathbf { k } , \quad \mathbf { b } = 2 \mathbf { i } + \mathbf { j } + 3 \mathbf { k } , \quad \mathbf { c } = \mathbf { i } + 5 \mathbf { j } + 4 \mathbf { k } , \quad \mathbf { d } = 3 \mathbf { i } + 6 \mathbf { j } + 5 \mathbf { k } , \quad \mathbf { g } = 3 \mathbf { i } + 4 \mathbf { j } + 5 \mathbf { k }$$ respectively.

1. The line through \(A\) and \(G\) meets the plane \(B C D\) at \(M\). Write down the vector equation of the line through \(A\) and \(G\) and hence show that the position vector of \(M\) is \(2 \mathbf { i } + 4 \mathbf { j } + 4 \mathbf { k }\).
2. Find the value of the ratio \(A G : A M\).
3. Find the position vector of the point \(P\) on the line through \(C\) and \(G\), such that \(\overrightarrow { C P } = \frac { 4 } { 3 } \overrightarrow { C G }\).
4. Verify that \(P\) lies in the plane \(A B D\).

2 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 \(8 x - 7 y + 10 z = 7\). Determine whether \(l\) lies in \(\Pi\), is parallel to \(\Pi\) without intersecting it, or intersects \(\Pi\) at one point.

6 Lines \(l _ { 1 }\) and \(l _ { 2 }\) have equations $$\frac { x - 3 } { 2 } = \frac { y - 4 } { - 1 } = \frac { z + 1 } { 1 } \quad \text { and } \quad \frac { x - 5 } { 4 } = \frac { y - 1 } { 3 } = \frac { z - 1 } { 2 }$$ respectively.

1. Find the equation of the plane \(\Pi _ { 1 }\) which contains \(l _ { 1 }\) and is parallel to \(l _ { 2 }\), giving your answer in the form r.n \(= p\).
2. Find the equation of the plane \(\Pi _ { 2 }\) which contains \(l _ { 2 }\) and is parallel to \(l _ { 1 }\), giving your answer in the form r.n \(= p\).
3. Find the distance between the planes \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\).
4. State the relationship between the answer to part (iii) and the lines \(l _ { 1 }\) and \(l _ { 2 }\).

2 In this question, \(L\) is the straight line with equation \(\mathbf { r } = \left( \begin{array} { r } 2 \\ 1 \\ - 1 \end{array} \right) + \lambda \left( \begin{array} { r } - 2 \\ 2 \\ 1 \end{array} \right)\), and \(\mathrm { g } ( x , y , z ) = \left( x y + z ^ { 2 } \right) \mathrm { e } ^ { x - 2 y }\).

1. Find \(\frac { \partial \mathrm { g } } { \partial x } , \frac { \partial \mathrm {~g} } { \partial y }\) and \(\frac { \partial \mathrm { g } } { \partial z }\).
2. Show that the normal to the surface \(\mathrm { g } ( x , y , z ) = 3\) at the point \(( 2,1 , - 1 )\) is the line \(L\). On the line \(L\), there are two points at which \(\mathrm { g } ( x , y , z ) = 0\).
3. Show that one of these points is \(\mathrm { P } ( 0,3,0 )\), and find the coordinates of the other point Q .
4. Show that, if \(x = - 2 \mu , y = 3 + 2 \mu , z = \mu\), and \(\mu\) is small, then $$\mathrm { g } ( x , y , z ) \approx - 6 \mu \mathrm { e } ^ { - 6 }$$ You are given that \(h\) is a small number.
5. There is a point on \(L\), close to P , at which \(\mathrm { g } ( x , y , z ) = h\). Show that this point is approximately $$\left( \frac { 1 } { 3 } \mathrm { e } ^ { 6 } h , 3 - \frac { 1 } { 3 } \mathrm { e } ^ { 6 } h , - \frac { 1 } { 6 } \mathrm { e } ^ { 6 } h \right)$$
6. Find the approximate coordinates of the point on \(L\), close to Q , at which \(\mathrm { g } ( x , y , z ) = h\).

5 The vector equations of two lines are $$\mathbf { r } = ( 5 \mathbf { i } - 2 \mathbf { j } - 2 \mathbf { k } ) + s ( 3 \mathbf { i } - 4 \mathbf { j } + 2 \mathbf { k } ) \quad \text { and } \quad \mathbf { r } = ( 2 \mathbf { i } - 2 \mathbf { j } + 7 \mathbf { k } ) + t ( 2 \mathbf { i } - \mathbf { j } - 5 \mathbf { k } ) .$$ Prove that the two lines are

1. perpendicular,
2. skew.

9 Lines \(L _ { 1 } , L _ { 2 }\) and \(L _ { 3 }\) have vector equations $$\begin{aligned} & L _ { 1 } : \mathbf { r } = ( 5 \mathbf { i } - \mathbf { j } - 2 \mathbf { k } ) + s ( - 6 \mathbf { i } + 8 \mathbf { j } - 2 \mathbf { k } ) , \\ & L _ { 2 } : \mathbf { r } = ( 3 \mathbf { i } - 8 \mathbf { j } ) + t ( \mathbf { i } + 3 \mathbf { j } + 2 \mathbf { k } ) , \\ & L _ { 3 } : \mathbf { r } = ( 2 \mathbf { i } + \mathbf { j } + 3 \mathbf { k } ) + u ( 3 \mathbf { i } + c \mathbf { j } + \mathbf { k } ) . \end{aligned}$$

1. Calculate the acute angle between \(L _ { 1 }\) and \(L _ { 2 }\).
2. Given that \(L _ { 1 }\) and \(L _ { 3 }\) are parallel, find the value of \(c\).
3. Given instead that \(L _ { 2 }\) and \(L _ { 3 }\) intersect, find the value of \(c\). 4

4 Relative to an origin \(O\), the points \(A\) and \(B\) have position vectors \(3 \mathbf { i } + 2 \mathbf { j } + 3 \mathbf { k }\) and \(\mathbf { i } + 3 \mathbf { j } + 4 \mathbf { k }\) respectively.

1. Find a vector equation of the line passing through \(A\) and \(B\).
2. Find the position vector of the point \(P\) on \(A B\) such that \(O P\) is perpendicular to \(A B\).

6 Two lines have equations $$\mathbf { r } = \left( \begin{array} { r } 1 \\ 0 \\ - 5 \end{array} \right) + t \left( \begin{array} { l } 2 \\ 3 \\ 4 \end{array} \right) \quad \text { and } \quad \mathbf { r } = \left( \begin{array} { r } 12 \\ 0 \\ 5 \end{array} \right) + s \left( \begin{array} { r } 1 \\ - 4 \\ - 2 \end{array} \right) .$$

1. Show that the lines intersect.
2. Find the angle between the lines.

7 The line \(L _ { 1 }\) passes through the point \(( 3,6,1 )\) and is parallel to the vector \(2 \mathbf { i } + 3 \mathbf { j } - \mathbf { k }\). The line \(L _ { 2 }\) passes through the point ( \(3 , - 1,4\) ) and is parallel to the vector \(\mathbf { i } - 2 \mathbf { j } + \mathbf { k }\).

1. Write down vector equations for the lines \(L _ { 1 }\) and \(L _ { 2 }\).
2. Prove that \(L _ { 1 }\) and \(L _ { 2 }\) intersect, and find the coordinates of their point of intersection.
3. Calculate the acute angle between the lines.

8 A pipeline is to be drilled under a river (see Fig. 8). With respect to axes Oxyz, with the \(x\)-axis pointing East, the \(y\)-axis North and the \(z\)-axis vertical, the pipeline is to consist of a straight section AB from the point \(\mathrm { A } ( 0 , - 40,0 )\) to the point \(\mathrm { B } ( 40,0 , - 20 )\) directly under the river, and another straight section BC . All lengths are in metres. \begin{figure}[h] \end{figure}

1. Calculate the distance AB . The section BC is to be drilled in the direction of the vector \(3 \mathbf { i } + 4 \mathbf { j } + \mathbf { k }\).
2. Find the angle ABC between the sections AB and BC . The section BC reaches ground level at the point \(\mathrm { C } ( a , b , 0 )\).
3. Write down a vector equation of the line BC . Hence find \(a\) and \(b\).
4. Show that the vector \(6 \mathbf { i } - 5 \mathbf { j } + 2 \mathbf { k }\) is perpendicular to the plane ABC . Hence find the cartesian equation of this plane.

7 A glass ornament OABCDEFG is a truncated pyramid on a rectangular base (see Fig. 7). All dimensions are in centimetres. \begin{figure}[h] \end{figure}

1. Write down the vectors \(\overrightarrow { \mathrm { CD } }\) and \(\overrightarrow { \mathrm { CB } }\).
2. Find the length of the edge CD.
3. Show that the vector \(4 \mathbf { i } + \mathbf { k }\) is perpendicular to the vectors \(\overrightarrow { \mathrm { CD } }\) and \(\overrightarrow { \mathrm { CB } }\). Hence find the cartesian equation of the plane BCDE .
4. Write down vector equations for the lines OG and AF . Show that they meet at the point P with coordinates (5, 10, 40). You may assume that the lines CD and BE also meet at the point P .
   The volume of a pyramid is \(\frac { 1 } { 3 } \times\) area of base × height.
5. Find the volumes of the pyramids POABC and PDEFG . Hence find the volume of the ornament.

5 Verify that the point \(( - 1,6,5 )\) lies on both the lines $$\mathbf { r } = \left( \begin{array} { r } 1 \\ 2 \\ - 1 \end{array} \right) + \lambda \left( \begin{array} { r } - 1 \\ 2 \\ 3 \end{array} \right) \quad \text { and } \quad \mathbf { r } = \left( \begin{array} { l } 0 \\ 6 \\ 3 \end{array} \right) + \mu \left( \begin{array} { l } - 1 \\ 0 \\ 1 \end{array} \right)$$

7 A straight pipeline AB passes through a mountain. With respect to axes \(\mathrm { O } x y z\), with \(\mathrm { O } x\) due East, \(\mathrm { O } y\) due North and \(\mathrm { O } z\) vertically upwards, A has coordinates \(( - 200,100,0 )\) and B has coordinates \(( 100,200,100 )\), where units are metres.

1. Verify that \(\overrightarrow { \mathrm { AB } } = \left( \begin{array} { l } 300 \\ 100 \\ 100 \end{array} \right)\) and find the length of the pipeline.
2. Write down a vector equation of the line AB , and calculate the angle it makes with the vertical. A thin flat layer of hard rock runs through the mountain. The equation of the plane containing this layer is \(x + 2 y + 3 z = 320\).
3. Find the coordinates of the point where the pipeline meets the layer of rock.
4. By calculating the angle between the line AB and the normal to the plane of the layer, find the angle at which the pipeline cuts through the layer.

9 A laser beam is aimed from a point ( \(12,10,10\) ) in the direction \(- 2 \mathbf { i } - 2 \mathbf { j } - 3 \mathbf { k }\) towards a plane surface.

1. Give the equation of the path of the laser beam in vector form. The points \(\mathrm { A } ( 1,1,1 ) , \mathrm { B } ( 1,4,2 )\) and \(\mathrm { C } ( 6,1,0 )\) lie on the plane.
2. Show that the vector \(3 \mathbf { i } - 5 \mathbf { j } + 15 \mathbf { k }\) is perpendicular to the plane and hence find the cartesian equation of the plane.
3. Find the coordinate of the point where the laser beam hits the surface of the plane.
4. Find the angle between the laser beam and the plane. \section*{Insert for question 6.} The graph of \(y = \tan x\) is given below.
   On this graph sketch the graph of \(y = \cot x\).
   Show clearly where your graph crosses the graph of \(y = \tan x\) and indicate the asymptotes. [4] \includegraphics[max width=\textwidth, alt={}, center]{23771896-942c-4a1d-ab95-6b6d3cc5643c-5_853_1555_703_262}

9 Beside a major route into a county town the authorities decide to build a large pyramid. Fig. 9.1 shows this pyramid, ABCDE O is the centre point of the horizontal base BCDE . A coordinate system is defined with O as the origin. The \(x\)-axis and \(y\)-axis are horizontal and the \(z\)-axis is vertical, as shown in Fig. 9.1 The vertices of the pyramid are $$A ( 0,0,6 ) , B ( - 4 , - 4,0 ) , C ( 4 , - 4,0 ) , D ( 4,4,0 ) \text { and } E ( - 4,4,0 ) .$$ \begin{figure}[h] \end{figure} The pyramid is supported by a vertical pole OA and there are also support rods from O to points on the triangular faces \(\mathrm { ABC } , \mathrm { ACD } , \mathrm { ADE }\) and AEB . One of the rods, ON , is shown in fig.9.2 which shows one quarter of the pyramid. \begin{figure}[h] \end{figure} M is the mid-point of the line BC .

1. Write down the coordinates of M.
2. Write down the vector \(\overrightarrow { \mathrm { AM } }\) and hence the coordinates of the point N which divides \(\overrightarrow { \mathrm { AM } }\) so that the ratio \(\mathrm { AN } : \mathrm { NM } = 2 : 1\).
3. Show that ON is perpendicular to both AM and BC .
4. Hence write down the equation of the plane ABC in its simplest form.
5. Find the angle between the face ABC and the ground.

2 Find where the line \(\mathbf { r } = \left( \begin{array} { l } 1 \\ 2 \\ 0 \end{array} \right) + \lambda \left( \begin{array} { l } 1 \\ 3 \\ 2 \end{array} \right)\) meets the plane \(2 x + 3 y - 4 z - 5 = 0\).

7. The line \(l _ { 1 }\) passes through the points \(A\) and \(B\) with position vectors ( \(3 \mathbf { i } + 6 \mathbf { j } - 8 \mathbf { k }\) ) and ( \(8 \mathbf { j } - 6 \mathbf { k }\) ) respectively, relative to a fixed origin.

1. Find a vector equation for \(l _ { 1 }\). The line \(l _ { 2 }\) has vector equation $$\mathbf { r } = ( - 2 \mathbf { i } + 10 \mathbf { j } + 6 \mathbf { k } ) + \mu ( 7 \mathbf { i } - 4 \mathbf { j } + 6 \mathbf { k } ) ,$$ where \(\mu\) is a scalar parameter.
2. Show that lines \(l _ { 1 }\) and \(l _ { 2 }\) intersect.
3. Find the coordinates of the point where \(l _ { 1 }\) and \(l _ { 2 }\) intersect. The point \(C\) lies on \(l _ { 2 }\) and is such that \(A C\) is perpendicular to \(A B\).
4. Find the position vector of \(C\).

7. Relative to a fixed origin, two lines have the equations
and $$\begin{aligned} & \mathbf { r } = \left( \begin{array} { c } 7 \\ 0 \\ - 3 \end{array} \right) + s \left( \begin{array} { c } 5 \\ 4 \\ - 2 \end{array} \right) \\ & \mathbf { r } = \left( \begin{array} { l } a \\ 6 \\ 3 \end{array} \right) + t \left( \begin{array} { c } - 5 \\ 14 \\ 2 \end{array} \right) , \end{aligned}$$ where \(a\) is a constant and \(s\) and \(t\) are scalar parameters.
Given that the two lines intersect,

1. find the position vector of their point of intersection,
2. find the value of \(a\). Given also that \(\theta\) is the acute angle between the lines,
3. find the value of \(\cos \theta\) in the form \(k \sqrt { 5 }\) where \(k\) is rational.

4. The line \(l _ { 1 }\) passes through the points \(P\) and \(Q\) with position vectors ( \(- \mathbf { i } - 8 \mathbf { j } + 3 \mathbf { k }\) ) and ( \(2 \mathbf { i } - 9 \mathbf { j } + \mathbf { k }\) ) respectively, relative to a fixed origin.

1. Find a vector equation for \(l _ { 1 }\). The line \(l _ { 2 }\) has the equation $$\mathbf { r } = ( 6 \mathbf { i } + a \mathbf { j } + b \mathbf { k } ) + t ( \mathbf { i } + 4 \mathbf { j } - \mathbf { k } )$$ and also passes through the point \(Q\).
2. Find the values of the constants \(a\) and \(b\).
3. Find, in degrees to 1 decimal place, the acute angle between lines \(l _ { 1 }\) and \(l _ { 2 }\).

8. The points \(A\) and \(B\) have coordinates \(( 3,9 , - 7 )\) and \(( 13 , - 6 , - 2 )\) respectively.

1. Find, in vector form, an equation for the line \(l\) which passes through \(A\) and \(B\).
2. Show that the point \(C\) with coordinates \(( 9,0 , - 4 )\) lies on \(l\). The point \(D\) is the point on \(l\) closest to the origin, \(O\).
3. Find the coordinates of \(D\).
4. Find the area of triangle \(O A B\) to 3 significant figures.

4 Show that the straight lines with equations \(\mathbf { r } = \begin{array} { r r r } 2 & + \lambda & 0 \\ 4 & & 1 \end{array}\) and \(\mathbf { r } = \quad + \mu \quad\) meet.
Find their point of intersection.

6

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