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

5 The line \(l _ { 1 }\) has equation \(\frac { x + 5 } { 1 } = \frac { y + 4 } { - 3 } = \frac { z - 3 } { 5 }\) The plane \(\Pi _ { 1 }\) has equation \(2 x + 3 y - 2 z = 6\)

1. Find the point of intersection of \(l _ { 1 }\) and \(\Pi _ { 1 }\) The line \(l _ { 2 }\) is the reflection of the line \(l _ { 1 }\) in the plane \(\Pi _ { 1 }\)
2. Show that a vector equation for the line \(l _ { 2 }\) is $$\mathbf { r } = \left( \begin{array} { r } - 7 \\ 2 \\ - 7 \end{array} \right) + \mu \left( \begin{array} { c } 10 \\ 6 \\ 2 \end{array} \right)$$ where \(\mu\) is a scalar parameter. The plane \(\Pi _ { 2 }\) contains the line \(l _ { 1 }\) and the line \(l _ { 2 }\)
3. Determine a vector equation for the line of intersection of \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\) The plane \(\Pi _ { 3 }\) has equation r. \(\left( \begin{array} { l } 1 \\ 1 \\ a \end{array} \right) = b\) where \(a\) and \(b\) are constants.
   Given that the planes \(\Pi _ { 1 } , \Pi _ { 2 }\) and \(\Pi _ { 3 }\) form a sheaf,
4. determine the value of \(a\) and the value of \(b\).

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

$$\mathbf { r } = \mathbf { i } - 2 \mathbf { j } + 3 \mathbf { k } + \lambda ( 2 \mathbf { i } + \mathbf { j } - 4 \mathbf { k } )$$ and the line \(l _ { 2 }\) has equation $$\mathbf { r } = 5 \mathbf { i } + p \mathbf { j } - 7 \mathbf { k } + \mu ( 6 \mathbf { i } + \mathbf { j } + 8 \mathbf { k } )$$ where \(\lambda\) and \(\mu\) are scalar parameters and \(p\) is a constant.
The plane \(\Pi\) contains \(l _ { 1 }\) and \(l _ { 2 }\)

1. Show that the vector \(3 \mathbf { i } - 10 \mathbf { j } - \mathbf { k }\) is perpendicular to \(\Pi\)
2. Hence determine a Cartesian equation of \(\Pi\)
3. Hence determine the value of \(p\) Given that
   • the lines \(l _ { 1 }\) and \(l _ { 2 }\) intersect at the point \(A\)
   • the point \(B\) has coordinates \(( 12 , - 11,6 )\)
   • determine, to the nearest degree, the acute angle between \(A B\) and \(\Pi\)

1. The line \(l _ { 1 }\) has equation \(\frac { x - 2 } { 4 } = \frac { y - 4 } { - 2 } = \frac { z + 6 } { 1 }\)

The plane \(\Pi\) has equation \(x - 2 y + z = 6\) The line \(l _ { 2 }\) is the reflection of the line \(l _ { 1 }\) in the plane \(\Pi\).
Find a vector equation of the line \(l _ { 2 }\)

6. $$\mathbf { M } = \left( \begin{array} { r r r } k & 5 & 7 \\ 1 & 1 & 1 \\ 2 & 1 & - 1 \end{array} \right) \quad \text { where } k \text { is a constant }$$

1. Given that \(k \neq 4\), find, in terms of \(k\), the inverse of the matrix \(\mathbf { M }\).
2. Find, in terms of \(p\), the coordinates of the point where the following planes intersect. $$\begin{array} { r } 2 x + 5 y + 7 z = 1 \\ x + y + z = p \\ 2 x + y - z = 2 \end{array}$$
3. 1. Find the value of \(q\) for which the following planes intersect in a straight line. $$\begin{array} { r } 4 x + 5 y + 7 z = 1 \\ x + y + z = q \\ 2 x + y - z = 2 \end{array}$$
   2. For this value of \(q\), determine a vector equation for the line of intersection.

1. With respect to a fixed origin \(O\), the line \(l\) has equation

$$( \mathbf { r } - ( 12 \mathbf { i } + 16 \mathbf { j } - 8 \mathbf { k } ) ) \times ( 9 \mathbf { i } + 6 \mathbf { j } + 2 \mathbf { k } ) = \mathbf { 0 }$$ The point \(A\) lies on \(l\) such that the direction cosines of \(\overrightarrow { O A }\) with respect to the \(\mathbf { i } , \mathbf { j }\) and \(\mathbf { k }\) axes are \(\frac { 3 } { 7 } , \beta\) and \(\gamma\). Determine the coordinates of the point \(A\).

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

The line \(l _ { 2 }\) has equation \(\mathbf { r } = \left( \begin{array} { c } 13 \\ 5 \\ 8 \end{array} \right) + \mu \left( \begin{array} { r } 1 \\ - 2 \\ 5 \end{array} \right)\) where \(\lambda\) and \(\mu\) are scalar parameters.
The lines \(l _ { 1 }\) and \(l _ { 2 }\) intersect at the point \(P\).

1. Determine the coordinates of \(P\). Given that the plane \(\Pi\) contains both \(l _ { 1 }\) and \(l _ { 2 }\)
2. determine a Cartesian equation for \(\Pi\).
   (ii) Determine a Cartesian equation for each of the two lines that
   • pass through \(( 0,0,0 )\)
   • make an angle of \(60 ^ { \circ }\) with the \(x\)-axis
   • make an angle of \(45 ^ { \circ }\) with the \(y\)-axis

3
1 \end{array} \right) + \lambda \left( \begin{array} { r } - 2
4
- 2 \end{array} \right)
& l _ { 2 } : \mathbf { r } = \left( \begin{array} { l }

2

1. For non-zero vectors \(\mathbf { x }\) and \(\mathbf { y }\), explain the geometrical significance of the statement \(\mathbf { x } \times \mathbf { y } = \mathbf { 0 }\).
2. The points \(P\) and \(Q\) have position vectors \(\mathbf { p } = 2 \mathbf { i } + 7 \mathbf { j } - \mathbf { k }\) and \(\mathbf { q } = \mathbf { i } + 3 \mathbf { j } + 4 \mathbf { k }\) respectively. Find, in the form \(( \mathbf { r } - \mathbf { a } ) \times \mathbf { d } = \mathbf { 0 }\), the equation of line \(P Q\).

4 The lines \(l _ { 1 }\) and \(l _ { 2 }\) have equations \(\frac { x - 7 } { 2 } = \frac { y - 1 } { - 1 } = \frac { z - 6 } { 3 }\) and \(\frac { x - 2 } { 1 } = \frac { y - 6 } { 2 } = \frac { z + 2 } { 1 }\) respectively.

1. Show that \(l _ { 1 }\) and \(l _ { 2 }\) intersect.
2. Find the cartesian equation of the plane that contains \(l _ { 1 }\) and \(l _ { 2 }\).

7. The lines \(l _ { 1 }\) and \(l _ { 2 }\) have equations $$\mathbf { r } = \left( \begin{array} { r } 1 \\ - 1 \\ 2 \end{array} \right) + \lambda \left( \begin{array} { r } - 1 \\ 3 \\ 4 \end{array} \right) \quad \text { and } \quad \mathbf { r } = \left( \begin{array} { r } \alpha \\ - 4 \\ 0 \end{array} \right) + \mu \left( \begin{array} { l } 0 \\ 3 \\ 2 \end{array} \right) .$$ 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 The quadrilateral \(A B C D\) has vertices \(A ( 2,1,3 ) , B ( 6,5,3 ) , C ( 6,1 , - 1 )\) and \(D ( 2 , - 3 , - 1 )\).
The line \(l _ { 1 }\) has vector equation \(\mathbf { r } = \left[ \begin{array} { r } 6 \\ 1 \\ - 1 \end{array} \right] + \lambda \left[ \begin{array} { l } 1 \\ 1 \\ 0 \end{array} \right]\).

1. 1. Find the vector \(\overrightarrow { A B }\).
   2. Show that the line \(A B\) is parallel to \(l _ { 1 }\).
   3. Verify that \(D\) lies on \(l _ { 1 }\).
2. The line \(l _ { 2 }\) passes through \(D ( 2 , - 3 , - 1 )\) and \(M ( 4,1,1 )\).
   1. Find the vector equation of \(l _ { 2 }\).
   2. Find the angle between \(l _ { 2 }\) and \(A C\).

9 The points \(A\) and \(B\) lie on the line \(l _ { 1 }\) and have coordinates \(( 2,5,1 )\) and \(( 4,1 , - 2 )\) respectively.

1. 1. Find the vector \(\overrightarrow { A B }\).
   2. Find a vector equation of the line \(l _ { 1 }\), with parameter \(\lambda\).
2. The line \(l _ { 2 }\) has equation \(\mathbf { r } = \left[ \begin{array} { r } 1 \\ - 3 \\ - 1 \end{array} \right] + \mu \left[ \begin{array} { r } 1 \\ 0 \\ - 2 \end{array} \right]\).
   1. Show that the point \(P ( - 2 , - 3,5 )\) lies on \(l _ { 2 }\).
   2. The point \(Q\) lies on \(l _ { 1 }\) and is such that \(P Q\) is perpendicular to \(l _ { 2 }\). Find the coordinates of \(Q\).

8 The points \(A\) and \(B\) have coordinates \(( 2,1 , - 1 )\) and \(( 3,1 , - 2 )\) respectively. The angle \(O B A\) is \(\theta\), where \(O\) is the origin.

1. 1. Find the vector \(\overrightarrow { A B }\).
   2. Show that \(\cos \theta = \frac { 5 } { 2 \sqrt { 7 } }\).
2. The point \(C\) is such that \(\overrightarrow { O C } = 2 \overrightarrow { O B }\). The line \(l\) is parallel to \(\overrightarrow { A B }\) and passes through the point \(C\). Find a vector equation of \(l\).
3. The point \(D\) lies on \(l\) such that angle \(O D C = 90 ^ { \circ }\). Find the coordinates of \(D\).

8 The points \(A , B\) and \(C\) have coordinates \(( 2 , - 1 , - 5 ) , ( 0,5 , - 9 )\) and \(( 9,2,3 )\) respectively. The line \(l\) has equation \(\mathbf { r } = \left[ \begin{array} { r } 2 \\ - 1 \\ - 5 \end{array} \right] + \lambda \left[ \begin{array} { r } 1 \\ - 3 \\ 2 \end{array} \right]\).

1. Verify that the point \(B\) lies on the line \(l\).
2. Find the vector \(\overrightarrow { B C }\).
3. The point \(D\) is such that \(\overrightarrow { A D } = 2 \overrightarrow { B C }\).
   1. Show that \(D\) has coordinates \(( 20 , - 7,19 )\).
   2. The point \(P\) lies on \(l\) where \(\lambda = p\). The line \(P D\) is perpendicular to \(l\). Find the value of \(p\).

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

1. Show that the distance between the points \(A\) and \(B\) is \(3 \sqrt { 3 }\).
2. The line \(A B\) makes an acute angle \(\theta\) with \(l\). Show that \(\cos \theta = \frac { 7 } { 9 }\).
3. The point \(P\) on the line \(l\) is where \(\lambda = p\).
   1. Show that $$\overrightarrow { A P } \cdot \left[ \begin{array} { r } 1 \\ - 1 \\ 1 \end{array} \right] = 7 + 3 p$$
   2. Hence find the coordinates of the foot of the perpendicular from the point \(A\) to the line \(l\).

7 The lines \(l _ { 1 }\) and \(l _ { 2 }\) have equations \(\mathbf { r } = \left[ \begin{array} { r } 8 \\ 6 \\ - 9 \end{array} \right] + \lambda \left[ \begin{array} { r } 3 \\ - 3 \\ - 1 \end{array} \right]\) and \(\mathbf { r } = \left[ \begin{array} { r } - 4 \\ 0 \\ 11 \end{array} \right] + \mu \left[ \begin{array} { r } 1 \\ 2 \\ - 3 \end{array} \right]\) respectively.

1. Show that \(l _ { 1 }\) and \(l _ { 2 }\) are perpendicular.
2. Show that \(l _ { 1 }\) and \(l _ { 2 }\) intersect and find the coordinates of the point of intersection, \(P\).
3. The point \(A ( - 4,0,11 )\) lies on \(l _ { 2 }\). The point \(B\) on \(l _ { 1 }\) is such that \(A P = B P\). Find the length of \(A B\).

7 The coordinates of the points \(A\) and \(B\) are ( \(3 , - 2,1\) ) and ( \(5,3,0\) ) respectively. The line \(l\) has equation \(\mathbf { r } = \left[ \begin{array} { l } 5 \\ 3 \\ 0 \end{array} \right] + \lambda \left[ \begin{array} { r } 1 \\ 0 \\ - 3 \end{array} \right]\).

1. Find the distance between \(A\) and \(B\).
2. Find the acute angle between the lines \(A B\) and \(l\). Give your answer to the nearest degree.
3. The points \(B\) and \(C\) lie on \(l\) such that the distance \(A C\) is equal to the distance \(A B\). Find the coordinates of \(C\).

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

1. Find the distance between the points \(A\) and \(B\).
2. Verify that \(B\) lies on \(l _ { 1 }\).
   (2 marks)
3. The line \(l _ { 2 }\) passes through \(A\) and has equation \(\mathbf { r } = \left[ \begin{array} { r } 3 \\ - 2 \\ 5 \end{array} \right] + \mu \left[ \begin{array} { r } - 1 \\ 3 \\ - 8 \end{array} \right]\). The lines \(l _ { 1 }\) and \(l _ { 2 }\) intersect at the point \(C\). Show that the points \(A , B\) and \(C\) form an isosceles triangle.
   (6 marks)

3 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 \(4 \mathbf { i } - 4 \mathbf { j } + 5 \mathbf { k }\).

1. Find an equation for \(L _ { 1 }\) in the form \(\mathbf { r } = \mathbf { a } + t \mathbf { b }\).
2. Prove that \(L _ { 1 }\) and \(L _ { 2 }\) are skew.

8 The upper and lower surfaces of a coal seam are modelled as planes ABC and DEF, as shown in Fig. 8. All dimensions are metres. \begin{figure}[h] \end{figure} Relative to axes \(\mathrm { O } x\) (due east), \(\mathrm { O } y\) (due north) and \(\mathrm { O } z\) (vertically upwards), the coordinates of the points are as follows.
A: (0, 0, -15)
B: (100, 0, -30)
C: (0, 100, -25)
D: (0, 0, -40)
E: (100, 0, -50)
F: (0, 100, -35)

1. Verify that the cartesian equation of the plane ABC is \(3 x + 2 y + 20 z + 300 = 0\).
2. Find the vectors \(\overrightarrow { \mathrm { DE } }\) and \(\overrightarrow { \mathrm { DF } }\). Show that the vector \(2 \mathbf { i } - \mathbf { j } + 20 \mathbf { k }\) is perpendicular to each of these vectors. Hence find the cartesian equation of the plane DEF .
3. By calculating the angle between their normal vectors, find the angle between the planes ABC and DEF. It is decided to drill down to the seam from a point \(\mathrm { R } ( 15,34,0 )\) in a line perpendicular to the upper surface of the seam. This line meets the plane ABC at the point S .
4. Write down a vector equation of the line RS. Calculate the coordinates of S.

1 The point A has coordinates \(( 2,5,4 )\) and the line BC has equation $$\mathbf { r } = \left( \begin{array} { c } 8 \\ 25 \\ 43 \end{array} \right) + \lambda \left( \begin{array} { c } 4 \\ 15 \\ 25 \end{array} \right)$$ You are given that \(\mathrm { AB } = \mathrm { AC } = 15\).

1. Show that the coordinates of one of the points B and C are (4, 10, 18). Find the coordinates of the other point. These points are B and C respectively.
2. Find the equation of the plane ABC in cartesian form.
3. Show that the plane containing the line BC and perpendicular to the plane ABC has equation \(5 y - 3 z + 4 = 0\). The point D has coordinates \(( 1,1,3 )\).
4. Show that \(| \overrightarrow { B C } \times \overrightarrow { A D } | = \sqrt { 7667 }\) and hence find the shortest distance between the lines \(B C\) and \(A D\).
5. Find the volume of the tetrahedron ABCD .

3.The lines \(L _ { 1 }\) and \(L _ { 2 }\) have the equations $$L _ { 1 } : \mathbf { r } = \left( \begin{array} { l } 1 \\ 0 \\ 9 \end{array} \right) + s \left( \begin{array} { l } 2 \\ p \\ 6 \end{array} \right) \quad \text { and } \quad L _ { 2 } : \mathbf { r } = \left( \begin{array} { r } - 15 \\ 12 \\ - 9 \end{array} \right) + t \left( \begin{array} { r } 4 \\ - 5 \\ 2 \end{array} \right)$$ where \(p\) is a constant.
The acute angle between \(L _ { 1 }\) and \(L _ { 2 }\) is \(\theta\) where \(\cos \theta = \frac { \sqrt { 5 } } { 3 }\)

1. Find the value of \(p\) . The line \(L _ { 3 }\) has equation \(\mathbf { r } = \left( \begin{array} { r } - 15 \\ 12 \\ - 9 \end{array} \right) + u \left( \begin{array} { r } 8 \\ - 6 \\ - 5 \end{array} \right)\) and the lines \(L _ { 3 }\) and \(L _ { 2 }\) intersect at the point \(A\) .
   The point \(B\) on \(L _ { 2 }\) has position vector \(\left( \begin{array} { r } 5 \\ - 13 \\ 1 \end{array} \right)\) and point \(C\) lies on \(L _ { 3 }\) such that \(A B D C\) is a rhombus.
2. Find the two possible position vectors of \(D\) .

15 Two submarines are travelling on different straight lines. The two lines are described by the equations $$\mathbf { r } = \left[ \begin{array} { c } 2 \\ - 1 \\ 4 \end{array} \right] + \lambda \left[ \begin{array} { c } 5 \\ 3 \\ - 2 \end{array} \right] \quad \text { and } \quad \frac { x - 5 } { 4 } = \frac { y } { 2 } = 4 - z$$ 15

1. 1. Show that the two lines intersect.
      [0pt] [3 marks]
      15
      1. (ii) Find the position vector of the point of intersection.
         15
   2. Tracey says that the submarines will collide because there is a common point on the two lines. Explain why Tracey is not necessarily correct. 15
   3. Calculate the acute angle between the lines $$\mathbf { r } = \left[ \begin{array} { c } 2 \\ - 1 \\ 4 \end{array} \right] + \lambda \left[ \begin{array} { c } 5 \\ 3 \\ - 2 \end{array} \right] \quad \text { and } \quad \frac { x - 5 } { 4 } = \frac { y } { 2 } = 4 - z$$ Give your angle to the nearest \(0.1 ^ { \circ }\)

7 The lines \(l _ { 1 }\) and \(l _ { 2 }\) have equations $$\begin{aligned} & l _ { 1 } : \mathbf { r } = \left[ \begin{array} { c } 3 \\ 1 \\ - 2 \end{array} \right] + \lambda \left[ \begin{array} { c } 3 \\ - 4 \\ 1 \end{array} \right] \\ & l _ { 2 } : \mathbf { r } = \left[ \begin{array} { c } - 12 \\ a \\ - 3 \end{array} \right] + \mu \left[ \begin{array} { c } 3 \\ 2 \\ - 1 \end{array} \right] \end{aligned}$$ 7

1. Show that the point \(P ( - 3,9 , - 4 )\) lies on \(l _ { 1 }\) 7
2. Show that \(l _ { 1 }\) is perpendicular to \(l _ { 2 }\) 7
3. Given that the lines \(l _ { 1 }\) and \(l _ { 2 }\) intersect, calculate the value of the constant \(a\) 7
4. Hence, find the coordinates of the point of intersection of \(l _ { 1 }\) and \(l _ { 2 }\)