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

4. Relative to a fixed origin \(O\), the point \(A\) has position vector \(4 \mathbf { i } + 8 \mathbf { j } - \mathbf { k }\), and the point \(B\) has position vector \(7 \mathbf { i } + 14 \mathbf { j } + 5 \mathbf { k }\).

1. Find the vector \(\overrightarrow { A B }\).
2. Calculate the cosine of \(\angle O A B\).
3. Show that, for all values of \(\lambda\), the point P with position vector \(\lambda \mathbf { i } + 2 \lambda \mathbf { j } + ( 2 \lambda - 9 ) \mathbf { k }\) lies on the line through \(A\) and \(B\).
4. Find the value of \(\lambda\) for which \(O P\) is perpendicular to \(A B\).
5. Hence find the coordinates of the foot of the perpendicular from \(O\) to \(A B\).

3. Relative to a fixed origin, \(O\), the line \(l\) has the equation $$\mathbf { r } = ( \mathbf { i } + p \mathbf { j } - 5 \mathbf { k } ) + \lambda ( 3 \mathbf { i } - \mathbf { j } + q \mathbf { k } ) ,$$ where \(p\) and \(q\) are constants and \(\lambda\) is a scalar parameter.
Given that the point \(A\) with coordinates \(( - 5,9 , - 9 )\) lies on \(l\),

1. find the values of \(p\) and \(q\),
2. show that the point \(B\) with coordinates \(( 25 , - 1,11 )\) also lies on \(l\). The point \(C\) lies on \(l\) and is such that \(O C\) is perpendicular to \(l\).
3. Find the coordinates of \(C\).
4. Find the ratio \(A C : C B\) 3. continued

5. 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 $$\mathbf { r } = ( 2 \mathbf { i } + 16 \mathbf { j } - \mathbf { k } ) + \mu ( - 5 \mathbf { i } + 3 \mathbf { j } ) ,$$ where \(\mu\) is a scalar parameter.

1. Find the position vector of \(C\). Given that 1 unit on each coordinate axis represents 200 metres,
2. find the distance, in kilometres, from the village at \(A\) to the junction at \(C\).
   5. continued

5. Relative to a fixed origin, the points \(A , B\) and \(C\) have position vectors ( \(2 \mathbf { i } - \mathbf { j } + 6 \mathbf { k }\) ), \(( 5 \mathbf { i } - 4 \mathbf { j } )\) and \(( 7 \mathbf { i } - 6 \mathbf { j } - 4 \mathbf { k } )\) respectively.

1. Show that \(A , B\) and \(C\) all lie on a single straight line.
2. Write down the ratio \(A B : B C\) The point \(D\) has position vector \(( 3 \mathbf { i } + \mathbf { j } + 4 \mathbf { k } )\).
3. Show that \(A D\) is perpendicular to \(B D\).
4. Find the exact area of triangle \(A B D\).
   5. continued

4. Relative to a fixed origin, two lines have the equations $$\begin{aligned} & \mathbf { r } = \left( \begin{array} { c } 7 \\ 0 \\ - 3 \end{array} \right) + \lambda \left( \begin{array} { c } 5 \\ 4 \\ - 2 \end{array} \right) \end{aligned}$$

& \mathbf { r } = \left( \begin{array} { l } a
6
3 \end{array} \right)

7
0
- 3 \end{array} \right) + \lambda \left( \begin{array} { c } 5
4
- 2 \end{array} \right)
& \mathbf { r } = \left( \begin{array} { l } a
6
3 \end{array} \right) + \mu \left( \begin{array} { c } - 5

2. Three forces \(\mathbf { F } _ { 1 } = ( 3 \mathbf { i } - \mathbf { j } + \mathbf { k } ) \mathrm { N } , \mathbf { F } _ { 2 } = ( 2 \mathbf { i } - \mathbf { k } ) \mathrm { N }\), and \(\mathbf { F } _ { 3 }\) act on a rigid body. The force \(\mathbf { F } _ { 1 }\) acts through the point with position vector \(( \mathbf { i } + 2 \mathbf { j } + \mathbf { k } ) \mathrm { m }\), the force \(\mathbf { F } _ { 2 }\) acts through the point with position vector \(( \mathbf { i } - 2 \mathbf { j } ) \mathrm { m }\) and the force \(\mathbf { F } _ { 3 }\) acts through the point with position vector \(( \mathbf { i } + \mathbf { j } + \mathbf { k } ) \mathrm { m }\). Given that the system \(\mathbf { F } _ { 1 } , \mathbf { F } _ { 2 }\) and \(\mathbf { F } _ { 3 }\) reduces to a couple \(\mathbf { G }\),

1. find \(\mathbf { G }\). The line of action of \(\mathbf { F } _ { 3 }\) is changed so that the system \(\mathbf { F } _ { 1 } , \mathbf { F } _ { 2 }\) and \(\mathbf { F } _ { 3 }\) now reduces to a couple \(( 6 \mathbf { i } + 8 \mathbf { j } + 2 \mathbf { k } ) \mathrm { N }\) m.
2. Find an equation of the new line of action of \(\mathbf { F } _ { 3 }\), giving your answer in the form \(\mathbf { r } = \mathbf { a } + t \mathbf { b }\), where \(\mathbf { a }\) and \(\mathbf { b }\) are constant vectors.

2

1. Show that the vector \(\mathbf { i } + 4 \mathbf { j } + 2 \mathbf { k }\) is parallel to the plane \(2 \mathrm { x } + \mathrm { y } - 3 \mathrm { z } = 10\).
2. Determine the acute angle between the planes \(2 x + y - 3 z = 10\) and \(x - y - 3 z = 3\).

13 The points A and B have coordinates \(( 4,0 , - 1 )\) and \(( 10,4 , - 3 )\) respectively. The planes \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\) have equations \(x - 2 y = 5\) and \(2 x + 3 y - z = - 4\) respectively.

1. Find the acute angle between the line AB and the plane \(\Pi _ { 1 }\).
2. Show that the line AB meets \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\) at the same point, whose coordinates should be specified.
3. 1. Find \(( \mathbf { i } - 2 \mathbf { j } ) \times ( 2 \mathbf { i } + 3 \mathbf { j } - \mathbf { k } )\).
   2. Hence find the acute angle between the planes \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\).
   3. Find the shortest distance between the point A and the line of intersection of the planes \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\).

11 The plane \(\Pi\) has equation \(2 x - y + 2 z = 4\). The point \(P\) has coordinates \(( 8,4,5 )\).

1. Calculate the shortest distance from P to \(\Pi\). The line \(L\) has equation \(\frac { x - 2 } { 3 } = \frac { y } { 2 } = \frac { z + 3 } { 4 }\).
2. Verify that P lies on L .
3. Find the coordinates of the point of intersection of L and \(\Pi\).
4. Determine the acute angle between L and \(\Pi\).
5. Use the results of parts (b), (c) and (d) to verify your answer to part (a).

15

1. Show that the three planes with equations $$\begin{aligned} x + \lambda y + 3 z & = - 12 \\ 2 x + y + 5 z & = - 11 \\ x - 2 y + 2 z & = - 9 \end{aligned}$$ where \(\lambda\) is a constant, meet at a unique point except for one value of \(\lambda\) which is to be determined.
2. In the case \(\lambda = - 2\), use matrices to find the point of intersection P of the planes, showing your method clearly. The line \(l\) has equation \(\frac { x - 1 } { 2 } = \frac { y - 1 } { - 1 } = \frac { z + 2 } { - 2 }\).
3. Find a vector equation of \(l\).
4. Find the shortest distance between the point P and \(l\).
5. 1. Show that \(l\) is parallel to the plane \(x - 2 y + 2 z = - 9\).
   2. Find the distance between \(l\) and the plane \(x - 2 y + 2 z = - 9\).

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

1. Find the vector equations of the lines \(A B\) and \(C D\).
2. Determine whether or not the lines \(A B\) and \(C D\) are perpendicular.

5. The points \(A\) and \(B\) have coordinates \(( 3,4 , - 2 )\) and \(( - 2,0,7 )\) respectively. The equation of the plane \(\Pi\) is given by \(2 x + 3 y + 3 z = 27\).

1. Show that the vector equation of the line \(A B\) may be expressed in the form $$\mathbf { r } = ( 3 - 5 \lambda ) \mathbf { i } + ( 4 - 4 \lambda ) \mathbf { j } + ( - 2 + 9 \lambda ) \mathbf { k }$$
2. Find the coordinates of the point of intersection of the line \(A B\) and the plane \(\Pi\).

1. The line \(l\) has equation

$$\frac { x + 2 } { 1 } = \frac { y - 5 } { - 1 } = \frac { z - 4 } { - 3 }$$ The plane \(\Pi\) has equation $$\mathbf { r } . ( \mathbf { i } - 2 \mathbf { j } + \mathbf { k } ) = - 7$$ Determine whether the line \(l\) intersects \(\Pi\) at a single point, or lies in \(\Pi\), or is parallel to \(\Pi\) without intersecting it.
(5)

1. A gas company maintains a straight pipeline that passes under a mountain.

The pipeline is modelled as a straight line and one side of the mountain is modelled as a plane. There are accessways from a control centre to two access points on the pipeline.
Modelling the control centre as the origin \(O\), the two access points on the pipeline have coordinates \(P ( - 300,400 , - 150 )\) and \(Q ( 300,300 , - 50 )\), where the units are metres.

1. Find a vector equation for the line \(P Q\), giving your answer in the form \(\mathbf { r } = \mathbf { a } + \lambda \mathbf { b }\), where \(\lambda\) is a scalar parameter. The equation of the plane modelling the side of the mountain is \(2 x + 3 y - 5 z = 300\) The company wants to create a new accessway from this side of the mountain to the pipeline. The accessway will consist of a tunnel of shortest possible length between the pipeline and the point \(M ( 100 , k , 100 )\) on this side of the mountain, where \(k\) is a constant.
2. Using the model, find
   1. the coordinates of the point at which this tunnel will meet the pipeline,
   2. the length of this tunnel. It is only practical to construct the new accessway if it will be significantly shorter than both of the existing accessways, \(O P\) and \(O Q\).
3. Determine whether the company should build the new accessway.
4. Suggest one limitation of the model.

1. All units in this question are in metres.

A lawn is modelled as a plane that contains the points \(L ( - 2 , - 3 , - 1 ) , M ( 6 , - 2,0 )\) and \(N ( 2,0,0 )\), relative to a fixed origin \(O\).

1. Determine a vector equation of the plane that models the lawn, giving your answer in the form \(\mathbf { r } = \mathbf { a } + \lambda \mathbf { b } + \mu \mathbf { c }\)
2. 1. Show that, according to the model, the lawn is perpendicular to the vector \(\left( \begin{array} { c } 1 \\ 2 \\ - 10 \end{array} \right)\)
   2. Hence determine a Cartesian equation of the plane that models the lawn. There are two posts set in the lawn.
      There is a washing line between the two posts.
      The washing line is modelled as a straight line through points at the top of each post with coordinates \(P ( - 10,8,2 )\) and \(Q ( 6,4,3 )\).
3. Determine a vector equation of the line that models the washing line.
4. State a limitation of one of the models. The point \(R ( 2,5,2.75 )\) lies on the washing line.
5. Determine, according to the model, the shortest distance from the point \(R\) to the lawn, giving your answer to the nearest cm. Given that the shortest distance from the point \(R\) to the lawn is actually 1.5 m ,
6. use your answer to part (e) to evaluate the model, explaining your reasoning.

1. The surface of a horizontal tennis court is modelled as part of a horizontal plane, with the origin on the ground at the centre of the court, and

• i and j are unit vectors directed across the width and length of the court respectively
• \(\quad \mathbf { k }\) is a unit vector directed vertically upwards
• units are metres

After being hit, a tennis ball, modelled as a particle, moves along the path with equation $$\mathbf { r } = \left( - 4.1 + 9 \lambda - 2.3 \lambda ^ { 2 } \right) \mathbf { i } + ( - 10.25 + 15 \lambda ) \mathbf { j } + \left( 0.84 + 0.8 \lambda - \lambda ^ { 2 } \right) \mathbf { k }$$ where \(\lambda\) is a scalar parameter with \(\lambda \geqslant 0\) Assuming that the tennis ball continues on this path until it hits the ground,

1. find the value of \(\lambda\) at the point where the ball hits the ground. The direction in which the tennis ball is moving at a general point on its path is given by $$( 9 - 4.6 \lambda ) \mathbf { i } + 15 \mathbf { j } + ( 0.8 - 2 \lambda ) \mathbf { k }$$
2. Write down the direction in which the tennis ball is moving as it hits the ground.
3. Hence find the acute angle at which the tennis ball hits the ground, giving your answer in degrees to one decimal place. The net of the tennis court lies in the plane \(\mathbf { r } . \mathbf { j } = 0\)
4. Find the position of the tennis ball at the point where it is in the same plane as the net. The maximum height above the court of the top of the net is 0.9 m .
   Modelling the top of the net as a horizontal straight line,
5. state whether the tennis ball will pass over the net according to the model, giving a reason for your answer. With reference to the model,
6. decide whether the tennis ball will actually pass over the net, giving a reason for your answer.

1. The line \(l _ { 1 }\) has equation \(\mathbf { r } = \left( \begin{array} { r } - 2 \\ 2 \\ 0 \end{array} \right) + \lambda \left( \begin{array} { l } 3 \\ 0 \\ 1 \end{array} \right)\) where \(\lambda\) is a scalar parameter.

The line \(l _ { 2 }\) is parallel to \(\left( \begin{array} { r } 1 \\ 2 \\ - 3 \end{array} \right)\)

1. Show that \(l _ { 1 }\) and \(l _ { 2 }\) are perpendicular. The plane \(\Pi\) contains the line \(l _ { 1 }\) and is perpendicular to \(\left( \begin{array} { r } 1 \\ 2 \\ - 3 \end{array} \right)\)
2. Determine a Cartesian equation of \(\Pi\)
3. Verify that the point \(A ( 3,1,1 )\) lies on \(\Pi\) Given that
   • the point of intersection of \(\Pi\) and \(l _ { 2 }\) has coordinates \(( 2,3,2 )\)
   • the point \(B ( p , q , r )\) lies on \(l _ { 2 }\)
   • the distance \(A B\) is \(2 \sqrt { 5 }\)
   • \(p , q\) and \(r\) are positive integers
   • determine the coordinates of \(B\).

1. The drainage system for a sports field consists of underground pipes.

This situation is modelled with respect to a fixed origin \(O\).
According to the model,

• the surface of the sports field is a plane with equation \(z = 0\)
• the pipes are straight lines
• one of the pipes, \(P _ { 1 }\), passes through the points \(A ( 3,4 , - 2 )\) and \(B ( - 2 , - 8 , - 3 )\)
• a different pipe, \(P _ { 2 }\), has equation \(\frac { x - 1 } { 2 } = \frac { y - 3 } { 4 } = \frac { z + 1 } { - 2 }\)
• the units are metres
  1. Determine a vector equation of the line representing the pipe \(P _ { 1 }\)
  2. Determine the coordinates of the point at which the pipe \(P _ { 1 }\) meets the surface of the playing field, according to the model.

Determine, according to the model,

the acute angle between pipes \(P _ { 1 }\) and \(P _ { 2 }\), giving your answer in degrees to 3 significant figures,

the shortest distance between pipes \(P _ { 1 }\) and \(P _ { 2 }\)

5. \begin{figure}[h] \end{figure} The points \(A ( 3,2 , - 4 ) , B ( 9 , - 4,2 ) , C ( - 6 , - 10,8 )\) and \(D ( - 4 , - 5,10 )\) are the vertices of a tetrahedron. The plane with equation \(z = 0\) cuts the tetrahedron into two pieces, one on each side of the plane. The edges \(A B , A C\) and \(A D\) of the tetrahedron intersect the plane at the points \(M , N\) and \(P\) respectively, as shown in Figure 1. Determine

1. the coordinates of the points \(M , N\) and \(P\),
2. the area of triangle \(M N P\),
3. the exact volume of the solid \(B C D P N M\).

4. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of a solid sculpture made of glass and concrete. The sculpture is modelled as a parallelepiped. The sculpture is made up of a concrete solid in the shape of a tetrahedron, shown shaded in Figure 1, whose vertices are \(\mathrm { O } ( 0,0,0 ) , \mathrm { A } ( 2,0,0 ) , \mathrm { B } ( 0,3,1 )\) and \(\mathrm { C } ( 1,1,2 )\), where the units are in metres. The rest of the solid parallelepiped is made of glass which is glued to the concrete tetrahedron.

1. Find the surface area of the glued face of the tetrahedron.
2. Find the volume of glass contained in this parallelepiped.
3. Give a reason why the volume of concrete predicted by this model may not be an accurate value for the volume of concrete that was used to make the sculpture. \section*{Q uestion 4 continued}

1. The line \(l _ { 1 }\) has equation

$$\frac { x - 1 } { 2 } = \frac { y + 1 } { - 1 } = \frac { z - 4 } { 3 }$$ The line \(l _ { 2 }\) has equation $$\mathbf { r } = \mathbf { i } + 3 \mathbf { k } + t ( \mathbf { i } - \mathbf { j } + 2 \mathbf { k } )$$ where \(t\) is a scalar parameter.

1. Show that \(l _ { 1 }\) and \(l _ { 2 }\) lie in the same plane.
2. Write down a vector equation for the plane containing \(l _ { 1 }\) and \(l _ { 2 }\)
3. Find, to the nearest degree, the acute angle between \(l _ { 1 }\) and \(l _ { 2 }\)