Line intersection verification

1. The line \(l _ { 1 }\) has vector equation \(\mathbf { r } = \left( \begin{array} { r } 5 \\ - 2 \\ 4 \end{array} \right) + \lambda \left( \begin{array} { r } 6 \\ 3 \\ - 1 \end{array} \right)\), where \(\lambda\) is a scalar parameter. The line \(l _ { 2 }\) has vector equation \(\mathbf { r } = \left( \begin{array} { r } 10 \\ 5 \\ - 3 \end{array} \right) + \mu \left( \begin{array} { l } 3 \\ 1 \\ 2 \end{array} \right)\), where \(\mu\) is a scalar parameter.

Justify, giving reasons in each case, whether the lines \(l _ { 1 }\) and \(l _ { 2 }\) are parallel, intersecting or skew.
(6)

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

1. Show that \(l _ { 1 }\) and \(l _ { 2 }\) do not meet. The point \(A\) is on \(l _ { 1 }\) where \(\lambda = 1\), and the point \(B\) is on \(l _ { 2 }\) where \(\mu = 2\).
2. Find the cosine of the acute angle between \(A B\) and \(l _ { 1 }\).

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.

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.

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

9 \end{array} \right) + t \left( \begin{array} { c } 2
- 3
1 \end{array} \right)$$ (ii) Show that lines \(l _ { 1 }\) and \(l _ { 2 }\) do not intersect.
(iii) Find the position vector of the point \(C\) on \(l _ { 2 }\) such that \(\angle A B C = 90 ^ { \circ }\).
8. \(f ( x ) = \frac { 5 - 8 x } { ( 1 + 2 x ) ( 1 - x ) ^ { 2 } }\).
(i) Express \(\mathrm { f } ( x )\) in partial fractions.
(ii) Find the series expansion of \(\mathrm { f } ( x )\) in ascending powers of \(x\) up to and including the term in \(x ^ { 3 }\), simplifying each coefficient.
(iii) State the set of values of \(x\) for which your expansion is valid.

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.

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 \(a\).

5 The lines \(l _ { 1 }\) and \(l _ { 2 }\) have equations $$\mathbf { r } = \left( \begin{array} { l } 4 \\ 6 \\ 4 \end{array} \right)$$

10 Lines \(l _ { 1 }\) and \(l _ { 2 }\) have vector equations $$\mathbf { r } = - \mathbf { i } + 2 \mathbf { j } + 7 \mathbf { k } + t ( 2 \mathbf { i } + 2 \mathbf { j } + \mathbf { k } ) \quad \text { and } \quad \mathbf { r } = 2 \mathbf { i } + 9 \mathbf { j } - 4 \mathbf { k } + s ( \mathbf { i } + 3 \mathbf { j } - 2 \mathbf { k } )$$ respectively. The point \(A\) has coordinates ( \(- 3,0,6\) ) relative to the origin \(O\).

1. Show that \(A\) lies on \(l _ { 1 }\) and that \(O A\) is perpendicular to \(l _ { 1 }\).
2. Show that the line through \(O\) and \(A\) intersects \(l _ { 2 }\).
3. Given that the point of intersection in part (ii) is \(B\), find the ratio \(| \overrightarrow { O A } | : | \overrightarrow { B A } |\). \section*{THERE ARE NO QUESTIONS WRITTEN ON THIS PAGE.}

3 Determine whether the lines whose equations are $$\mathbf { r } = ( 1 + 2 \lambda ) \mathbf { i } - \lambda \mathbf { j } + ( 3 + 5 \lambda ) \mathbf { k } \text { and } \mathbf { r } = ( \mu - 1 ) \mathbf { i } + ( 5 - \mu ) \mathbf { j } + ( 2 - 5 \mu ) \mathbf { k }$$ are parallel, intersect or are skew.

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.

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.

9 Two straight lines have equations $$\mathbf { r } = \left( \begin{array} { r } 16 \\ 2 \\ 3 \end{array} \right) + \lambda \left( \begin{array} { r } 3 \\ 2 \\ - 1 \end{array} \right) \quad \text { and } \quad \mathbf { r } = \left( \begin{array} { r } - 3 \\ 8 \\ 12 \end{array} \right) + \mu \left( \begin{array} { r } 5 \\ - 6 \\ - 3 \end{array} \right) .$$ Show that the two lines intersect and find the coordinates of their point of intersection.

9 Determine whether the lines whose equations are $$\mathbf { r } = ( 4 + 2 \mu ) \mathbf { i } + ( 7 + 3 \mu ) \mathbf { j } + ( 3 + 7 \mu ) \mathbf { k } \quad \text { and } \quad \mathbf { r } = ( 35 - 5 \lambda ) \mathbf { i } + ( 6 + 2 \lambda ) \mathbf { j } + ( 14 + 3 \lambda ) \mathbf { k }$$ intersect, are parallel or are skew.

7 With respect to the origin \(O\), the points \(A\) and \(B\) have position vectors \(\mathbf { a }\) and \(\mathbf { b }\) respectively, where \(\mathbf { a } = 2 \mathbf { i } - 3 \mathbf { j } + 6 \mathbf { k }\) and \(\mathbf { b } = - \mathbf { i } - \mathbf { j } + 2 \mathbf { k }\). The lines \(L _ { 1 }\) and \(L _ { 2 }\) have the vector equations $$\mathbf { r } = \mathbf { a } + \lambda \mathbf { b } , \quad \mathbf { r } = 2 \mathbf { b } + \mu \mathbf { a }$$ respectively.

1. Determine whether or not \(L _ { 1 }\) and \(L _ { 2 }\) intersect.
2. Find the acute angle between the directions of \(L _ { 1 }\) and \(L _ { 2 }\). The point \(C\) has position vector \(\mathbf { c } = p \mathbf { i } + \mathbf { j } + r \mathbf { k }\).
3. Given that \(O C\) is perpendicular to the triangle \(O A B\), determine \(p\) and \(r\).
4. Determine the volume of the tetrahedron \(O A B C\).