Line intersection (vector form)

11. With respect to a fixed origin \(O\) the lines \(l _ { 1 }\) and \(l _ { 2 }\) are given by the equations $$l _ { 1 } : \mathbf { r } = \left( \begin{array} { r }

17 \end{array} \right) + \lambda \left( \begin{array} { c } - 2
1
- 4 \end{array} \right) \quad l _ { 2 } : \quad \mathbf { r } = \left( \begin{array} { c } - 5
11
p \end{array} \right) + \mu \left( \begin{array} { l } q
2
2 \end{array} \right)$$ where \(\lambda\) and \(\mu\) are parameters and \(p\) and \(q\) are constants. Given that \(l _ { 1 }\) and \(l _ { 2 }\) are perpendicular,

1. show that \(q = - 3\). Given further that \(l _ { 1 }\) and \(l _ { 2 }\) intersect, find
2. the value of \(p\),
3. the coordinates of the point of intersection. The point \(A\) lies on \(l _ { 1 }\) and has position vector \(\left( \begin{array} { c } 9 \\ 3 \\ 13 \end{array} \right)\). The point \(C\) lies on \(l _ { 2 }\).\\ Given that a circle, with centre \(C\), cuts the line \(l _ { 1 }\) at the points \(A\) and \(B\),
4. find the position vector of \(B\).\\ 5. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{a5579938-e202-4543-8513-6483ede49850-09_696_686_196_626} \captionsetup{labelformat=empty} \caption{Figure 2}
   \end{figure} A container is made in the shape of a hollow inverted right circular cone. The height of the container is 24 cm and the radius is 16 cm , as shown in Figure 2. Water is flowing into the container. When the height of water is \(h \mathrm {~cm}\), the surface of the water has radius \(r \mathrm {~cm}\) and the volume of water is \(V \mathrm {~cm} ^ { 3 }\).
5. Show that \(V = \frac { 4 \pi h ^ { 3 } } { 27 }\).\\[0pt] [The volume \(V\) of a right circular cone with vertical height \(h\) and base radius \(r\) is given by the formula \(V = \frac { 1 } { 3 } \pi r ^ { 2 } h\).] Water flows into the container at a rate of \(8 \mathrm {~cm} ^ { 3 } \mathrm {~s} ^ { - 1 }\).
6. Find, in terms of \(\pi\), the rate of change of \(h\) when \(h = 12\). 6. (a) Find \(\int \tan ^ { 2 } x \mathrm {~d} x\).
7. Use integration by parts to find \(\int \frac { 1 } { x ^ { 3 } } \ln x \mathrm {~d} x\).
8. Use the substitution \(u = 1 + e ^ { x }\) to show that $$\int \frac { \mathrm { e } ^ { 3 x } } { 1 + \mathrm { e } ^ { x } } \mathrm {~d} x = \frac { 1 } { 2 } \mathrm { e } ^ { 2 x } - \mathrm { e } ^ { x } + \ln \left( 1 + \mathrm { e } ^ { x } \right) + k$$ where \(k\) is a constant.\\ 7. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{a5579938-e202-4543-8513-6483ede49850-13_511_714_237_612} \captionsetup{labelformat=empty} \caption{Figure 3}
   \end{figure} The curve \(C\) shown in Figure 3 has parametric equations $$x = t ^ { 3 } - 8 t , \quad y = t ^ { 2 }$$ where \(t\) is a parameter. Given that the point \(A\) has parameter \(t = - 1\),
9. find the coordinates of \(A\). The line \(l\) is the tangent to \(C\) at \(A\).
10. Show that an equation for \(l\) is \(2 x - 5 y - 9 = 0\). The line \(l\) also intersects the curve at the point \(B\).
11. Find the coordinates of \(B\).

6. With respect to a fixed origin, the point \(A\) with position vector \(\mathbf { i } + 2 \mathbf { j } + 3 \mathbf { k }\) lies on the line \(l _ { 1 }\) with equation $$\mathbf { r } = \left( \begin{array} { l } 1 \\ 2 \\ 3 \end{array} \right) + \lambda \left( \begin{array} { r } 0 \\ 2 \\ - 1 \end{array} \right) , \quad \text { where } \lambda \text { is a scalar parameter, }$$ and the point \(B\) with position vector \(4 \mathbf { i } + p \mathbf { j } + 3 \mathbf { k }\), where \(p\) is a constant, lies on the line \(l _ { 2 }\) with equation $$\mathbf { r } = \left( \begin{array} { l }

1. With respect to a fixed origin \(O\), the lines \(l _ { 1 }\) and \(l _ { 2 }\) are given by the equations

$$l _ { 1 } : \mathbf { r } = \left( \begin{array} { r }

5. The vector equations of two straight lines are $$\begin{aligned} & \mathbf { r } = 5 \mathbf { i } + 3 \mathbf { j } - 2 \mathbf { k } + \lambda ( \mathbf { i } - 2 \mathbf { j } + 2 \mathbf { k } ) \quad \text { and } \\ & \mathbf { r } = 2 \mathbf { i } - 11 \mathbf { j } + a \mathbf { k } + \mu ( - 3 \mathbf { i } - 4 \mathbf { j } + 5 \mathbf { k } ) . \end{aligned}$$ Given that the two lines intersect, find

1. the coordinates of the point of intersection,
2. the value of the constant \(a\),
3. the acute angle between the two lines.

9 Two lines have vector equations $$\mathbf { r } = \left( \begin{array} { r } 4 \\ 2 \\ - 6 \end{array} \right) + t \left( \begin{array} { r } - 8 \\ 1 \\ - 2 \end{array} \right) \quad \text { and } \quad \mathbf { r } = \left( \begin{array} { r } - 2 \\ a \\ - 2 \end{array} \right) + s \left( \begin{array} { r } - 9 \\ 2 \\ - 5 \end{array} \right) ,$$ where \(a\) is a constant.

1. Calculate the acute angle between the lines.
2. Given that these two lines intersect, find \(a\) and the point of intersection.

7 Two lines have vector equations $$\mathbf { r } = \mathbf { i } - 2 \mathbf { j } + 4 \mathbf { k } + \lambda ( 3 \mathbf { i } + \mathbf { j } + a \mathbf { k } ) \quad \text { and } \quad \mathbf { r } = - 8 \mathbf { i } + 2 \mathbf { j } + 3 \mathbf { k } + \mu ( \mathbf { i } - 2 \mathbf { j } - \mathbf { k } ) ,$$ where \(a\) is a constant.

1. Given that the lines are skew, find the value that \(a\) cannot take.
2. Given instead that the lines intersect, find the point of intersection.

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.

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. Relative to a fixed origin, two lines have the equations $$\mathbf { r } = ( 7 \mathbf { j } - 4 \mathbf { k } ) + s ( 4 \mathbf { i } - 3 \mathbf { j } + \mathbf { k } )$$ and $$\mathbf { r } = ( - 7 \mathbf { i } + \mathbf { j } + 8 \mathbf { k } ) + t ( - 3 \mathbf { i } + 2 \mathbf { k } )$$ where \(s\) and \(t\) are scalar parameters.

1. Show that the two lines intersect and find the position vector of the point where they meet.
2. Find, in degrees to 1 decimal place, the acute angle between the lines.

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 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.

7

1. Show that the straight line with equation \(\mathbf { r } = \left( \begin{array} { r } 2 \\ - 3 \\ 5 \end{array} \right) + t \left( \begin{array} { r } 1 \\ 4 \\ - 2 \end{array} \right)\) meets the line passing through ( \(9,7,5\) ) and ( \(7,8,2\) ), and find the point of intersection of these lines.
2. Find the acute angle between these lines.

4 The equations of two lines are $$\mathbf { r } = \mathbf { i } + 2 \mathbf { j } + \lambda ( 2 \mathbf { i } + \mathbf { j } + 3 \mathbf { k } ) \text { and } \mathbf { r } = 6 \mathbf { i } + 8 \mathbf { j } + \mathbf { k } + \mu ( \mathbf { i } + 4 \mathbf { j } - 5 \mathbf { k } ) .$$

1. Show that these lines meet, and find the coordinates of the point of intersection.
2. Find the acute angle between these lines.

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. 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.

7 Show that the straight lines with equations \(\mathbf { r } = \left( \begin{array} { l } 4 \\ 2 \\ 4 \end{array} \right) + \lambda \left( \begin{array} { l } 3 \\ 0 \\ 1 \end{array} \right)\) and \(\mathbf { r } = \left( \begin{array} { r } - 1 \\ 4 \\ 9 \end{array} \right) + \mu \left( \begin{array} { r } - 1 \\ 1 \\ 3 \end{array} \right)\) meet.
Find their point of intersection.

5 The points \(A\) and \(B\) have coordinates \(( 5,1 , - 2 )\) and \(( 4 , - 1,3 )\) respectively.
The line \(l\) has equation \(\mathbf { r } = \left[ \begin{array} { r } - 8 \\ 5 \\ - 6 \end{array} \right] + \mu \left[ \begin{array} { r } 5 \\ 0 \\ - 2 \end{array} \right]\).

1. Find a vector equation of the line that passes through \(A\) and \(B\).
2. 1. Show that the line that passes through \(A\) and \(B\) intersects the line \(l\), and find the coordinates of the point of intersection, \(P\).
   2. The point \(C\) lies on \(l\) such that triangle \(P B C\) has a right angle at \(B\). Find the coordinates of \(C\).

5. Relative to a fixed origin, two lines have the equations $$\begin{aligned} & \mathbf { r } = \left( \begin{array} { l } 4 \\ 1 \\ 1 \end{array} \right) + s \left( \begin{array} { l } 1 \\ 4 \\ 5 \end{array} \right) \\ & \mathbf { r } = \left( \begin{array} { c } - 3 \\ 1 \\ - 6 \end{array} \right) + t \left( \begin{array} { l } 3 \\ a \\ b \end{array} \right) \end{aligned}$$ and
where \(a\) and \(b\) are constants and \(s\) and \(t\) are scalar parameters.
Given that the two lines are perpendicular,

1. find a linear relationship between \(a\) and \(b\). Given also that the two lines intersect,
2. find the values of \(a\) and \(b\),
3. find the coordinates of the point where they intersect.
   5. continued

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 $$\mathbf { r } = ( 6 \mathbf { i } + a \mathbf { j } + b \mathbf { k } ) + \mu ( \mathbf { i } + 4 \mathbf { j } - \mathbf { k } )$$ 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 \(l _ { 2 }\).
   4. 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

3 Two lines, \(l _ { 1 }\) and \(l _ { 2 }\), have the following equations. $$\begin{aligned} & l _ { 1 } : \mathbf { r } = \left( \begin{array} { c } - 11
10
3 \end{array} \right) + \lambda \left( \begin{array} { c } 2
- 2
1 \end{array} \right)
& l _ { 2 } : \mathbf { r } = \left( \begin{array} { l } 5
2

5

1. Find the cartesian equation of the plane through the point ( \(2 , - 1,4\) ) with normal vector $$\mathbf { n } = \left( \begin{array} { r } 1 \\ - 1 \\ 2 \end{array} \right)$$
2. Find the coordinates of the point of intersection of this plane and the straight line with equation $$\mathbf { r } = \left( \begin{array} { r } 7 \\ 12 \\ 9 \end{array} \right) + \lambda \left( \begin{array} { l } 1 \\ 3 \\ 2 \end{array} \right)$$