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

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 \(C\).

9 Two lines have equations $$\mathbf { r } = 3 \mathbf { i } + 5 \mathbf { j } - \mathbf { k } + \lambda ( 2 \mathbf { i } + \mathbf { j } + \mathbf { k } ) \text { and } \mathbf { r } = 4 \mathbf { i } + 10 \mathbf { j } + 19 \mathbf { k } + \mu ( \mathbf { i } - \mathbf { j } + \alpha \mathbf { k } ) ,$$ where \(\alpha\) is a constant.
Find the value of \(\alpha\) in each of the following cases.

1. The lines intersect at the point (7,7,1).
2. The angle between their directions is \(60 ^ { \circ }\).

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. Write down normal vectors to the planes \(2 x - y + z = 2\) and \(x - z = 1\).
   Hence find the acute angle between the planes.
2. Write down a vector equation of the line through \(( 2,0,1 )\) perpendicular to the plane \(2 x - y + z = 2\). Find the point of intersection of this line with the plane.

4 The points A , B and C have coordinates \(( 1,3 , - 2 ) , ( - 1,2 , - 3 )\) and \(( 0 , - 8,1 )\) respectively.

1. Find the vectors \(\overrightarrow { \mathrm { AB } }\) and \(\overrightarrow { \mathrm { AC } }\).
2. Show that the vector \(2 \mathbf { i } - \mathbf { j } - 3 \mathbf { k }\) is perpendicular to the plane ABC . Hence find the equation of the plane ABC .

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

1. Find the point of intersection of the line \(\mathbf { r } = \left( \begin{array} { r } - 8 \\ - 2 \\ 6 \end{array} \right) + \lambda \left( \begin{array} { r } - 3 \\ 0 \\ 1 \end{array} \right)\) and the plane \(2 x - 3 y + z = 11\).
2. Find the acute angle between the line and the normal to the plane. Section B (36 marks)

11 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.

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 line \(l _ { 1 }\) passes through the point \(A\) whose position vector is \(3 \mathbf { i } + \mathbf { j } + 2 \mathbf { k }\) and is parallel to the vector \(\mathbf { i } + \mathbf { j }\). The line \(l _ { 2 }\) passes through the point \(B\) whose position vector is \(- \mathbf { i } - \mathbf { k }\) and is parallel to the vector \(\mathbf { j } + 2 \mathbf { k }\). The point \(P\) is on \(l _ { 1 }\) and the point \(Q\) is on \(l _ { 2 }\) and \(P Q\) is perpendicular to both \(l _ { 1 }\) and \(l _ { 2 }\).

1. Find the length of \(P Q\).
2. Find the position vector of \(Q\).
3. Show that the perpendicular distance from \(Q\) to the plane containing \(A B\) and the line \(l _ { 1 }\) is \(\sqrt { } 3\).

6 The line \(l _ { 1 }\) passes through the point with position vector \(8 \mathbf { i } + 8 \mathbf { j } - 7 \mathbf { k }\) and is parallel to the vector \(4 \mathbf { i } + 3 \mathbf { j }\). The line \(l _ { 2 }\) passes through the point with position vector \(7 \mathbf { i } - 2 \mathbf { j } + 4 \mathbf { k }\) and is parallel to the vector \(4 \mathbf { i } - \mathbf { k }\). 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 \(l _ { 2 }\). In either order,

1. show that \(P Q = 13\),
2. find the position vectors of \(P\) and \(Q\).

10 The lines \(l _ { 1 }\) and \(l _ { 2 }\) have equations $$l _ { 1 } : \mathbf { r } = 6 \mathbf { i } + 5 \mathbf { j } + 4 \mathbf { k } + \lambda ( \mathbf { i } + \mathbf { j } + \mathbf { k } ) \quad \text { and } \quad l _ { 2 } : \mathbf { r } = 6 \mathbf { i } + 5 \mathbf { j } + 4 \mathbf { k } + \mu ( 4 \mathbf { i } + 6 \mathbf { j } + \mathbf { k } ) .$$ Find a cartesian equation of the plane \(\Pi\) containing \(l _ { 1 }\) and \(l _ { 2 }\). Find the position vector of the foot of the perpendicular from the point with position vector \(\mathbf { i } + 10 \mathbf { j } + 3 \mathbf { k }\) to \(\Pi\). The line \(l _ { 3 }\) has equation \(\mathbf { r } = \mathbf { i } + 10 \mathbf { j } + 3 \mathbf { k } + v ( 2 \mathbf { i } - 3 \mathbf { j } + \mathbf { k } )\). Find the shortest distance between \(l _ { 1 }\) and \(l _ { 3 }\).

The line \(l _ { 1 }\) passes through the point \(A\) whose position vector is \(4 \mathbf { i } + 7 \mathbf { j } - \mathbf { k }\) and is parallel to the vector \(3 \mathbf { i } + 2 \mathbf { j } - \mathbf { k }\). The line \(l _ { 2 }\) passes through the point \(B\) whose position vector is \(\mathbf { i } + 7 \mathbf { j } + 11 \mathbf { k }\) and is parallel to the vector \(\mathbf { i } - 6 \mathbf { j } - 2 \mathbf { k }\). The points \(P\) on \(l _ { 1 }\) and \(Q\) on \(l _ { 2 }\) are such that \(P Q\) is perpendicular to both \(l _ { 1 }\) and \(l _ { 2 }\). Find the position vectors of \(P\) and \(Q\). Find the shortest distance between the line through \(A\) and \(B\) and the line through \(P\) and \(Q\), giving your answer correct to 3 significant figures.

11 The line \(l _ { 1 }\) passes through the points \(A ( 2,3 , - 5 )\) and \(B ( 8,7 , - 13 )\). The line \(l _ { 2 }\) passes through the points \(C ( - 2,1,8 )\) and \(D ( 3 , - 1,4 )\). Find the shortest distance between the lines \(l _ { 1 }\) and \(l _ { 2 }\). The plane \(\Pi _ { 1 }\) passes through the points \(A , B\) and \(D\). The plane \(\Pi _ { 2 }\) passes though the points \(A , C\) and \(D\). Find the acute angle between \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\), giving your answer in degrees.

1 The vectors \(\mathbf { a } , \mathbf { b } , \mathbf { c }\) and \(\mathbf { d }\) in \(\mathbb { R } ^ { 3 }\) are given by $$\mathbf { a } = \left( \begin{array} { r } 2 \\ - 1 \\ 1 \end{array} \right)$$

, \quad \mathbf { b } = \left( \begin{array} { l } 1
1
1 \end{array} \right) , \quad \mathbf { c } = \left( \begin{array} { r } 0
1
- 1 \end{array} \right) \quad \text { and } \quad \mathbf { d } = \left( \begin{array} { r } 2
- 1
1 \end{array} \right)

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.
   }

8 Find a cartesian equation of the plane \(\Pi _ { 1 }\) passing through the points with coordinates \(( 2 , - 1,3 )\), \(( 4,2 , - 5 )\) and \(( - 1,3 , - 2 )\). The plane \(\Pi _ { 2 }\) has cartesian equation \(3 x - y + 2 z = 5\). Find the acute angle between \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\). Find a vector equation of the line of intersection of the planes \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\).

The position vectors of the points \(A , B , C , D\) are $$\mathbf { i } + \mathbf { j } + 3 \mathbf { k } , \quad 3 \mathbf { i } - \mathbf { j } + 5 \mathbf { k } , \quad 3 \mathbf { i } - \mathbf { j } + \mathbf { k } , \quad 5 \mathbf { i } - 5 \mathbf { j } + \alpha \mathbf { k } ,$$ respectively, where \(\alpha\) is a positive integer. It is given that the shortest distance between the line \(A B\) and the line \(C D\) is equal to \(2 \sqrt { } 2\).

1. Show that the possible values of \(\alpha\) are 3 and 5 .
2. Using \(\alpha = 3\), find the shortest distance of the point \(D\) from the line \(A C\), giving your answer correct to 3 significant figures.
3. Using \(\alpha = 3\), find the acute angle between the planes \(A B C\) and \(A B D\), giving your answer in degrees.
   {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 \(l _ { 2 }\).

3 The lines \(l _ { 1 }\) and \(l _ { 2 }\) have equations \(\mathbf { r } = 6 \mathbf { i } + 2 \mathbf { j } + 7 \mathbf { k } + \lambda ( \mathbf { i } + \mathbf { j } )\) and \(\mathbf { r } = 4 \mathbf { i } + 4 \mathbf { j } + \mu ( - 6 \mathbf { j } + \mathbf { k } )\) 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 vectors of \(P\) and \(Q\).

11 The line \(l _ { 1 }\) passes through the point \(A\), whose position vector is \(3 \mathbf { i } - 5 \mathbf { j } - 4 \mathbf { k }\), and is parallel to the vector \(3 \mathbf { i } + 4 \mathbf { j } + 2 \mathbf { k }\). The line \(l _ { 2 }\) passes through the point \(B\), whose position vector is \(2 \mathbf { i } + 3 \mathbf { j } + 5 \mathbf { k }\), and is parallel to the vector \(\mathbf { i } - \mathbf { j } - 4 \mathbf { k }\). 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 \(l _ { 2 }\). The plane \(\Pi _ { 1 }\) contains \(P Q\) and \(l _ { 1 }\), and the plane \(\Pi _ { 2 }\) contains \(P Q\) and \(l _ { 2 }\).

1. Find the length of \(P Q\).
2. Find a vector perpendicular to \(\Pi _ { 1 }\).
3. Find the perpendicular distance from \(B\) to \(\Pi _ { 1 }\).
4. Find the angle between \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\).

9 Find a cartesian equation of the plane \(\Pi\) containing the lines $$\mathbf { r } = 3 \mathbf { i } + \mathbf { k } + s ( 2 \mathbf { i } + \mathbf { j } - \mathbf { k } ) \quad \text { and } \quad \mathbf { r } = 3 \mathbf { i } - 7 \mathbf { j } + 10 \mathbf { k } + t ( \mathbf { i } - 3 \mathbf { j } + 4 \mathbf { k } )$$ The line \(l\) passes through the point \(P\) with position vector \(6 \mathbf { i } - 2 \mathbf { j } + \mathbf { k }\) and is parallel to the vector \(2 \mathbf { i } + \mathbf { j } - 4 \mathbf { k }\). Find

1. the position vector of the point where \(l\) meets \(\Pi\),
2. the perpendicular distance from \(P\) to \(\Pi\),
3. the acute angle between \(l\) and \(\Pi\).

9 The plane \(\Pi\) has equation $$\mathbf { r } = 2 \mathbf { i } + 3 \mathbf { j } - \mathbf { k } + \lambda ( \mathbf { i } - 2 \mathbf { j } + 2 \mathbf { k } ) + \mu ( 3 \mathbf { i } + \mathbf { j } - 2 \mathbf { k } )$$ The line \(l\), which does not lie in \(\Pi\), has equation $$\mathbf { r } = 3 \mathbf { i } + 6 \mathbf { j } + 12 \mathbf { k } + t ( 8 \mathbf { i } + 5 \mathbf { j } - 8 \mathbf { k } )$$ Show that \(l\) is parallel to \(\Pi\). Find the position vector of the point at which the line with equation \(\mathbf { r } = 5 \mathbf { i } - 4 \mathbf { j } + 7 \mathbf { k } + s ( 2 \mathbf { i } - \mathbf { j } + \mathbf { k } )\) meets \(\Pi\). Find the perpendicular distance from the point with position vector \(9 \mathbf { i } + 11 \mathbf { j } + 2 \mathbf { k }\) to \(l\).