Angle between two lines

7. With respect to a fixed origin \(O\), the lines \(l _ { 1 }\) and \(l _ { 2 }\) are given by the equations $$\begin{aligned} & l _ { 1 } : \mathbf { r } = ( 13 \mathbf { i } + 15 \mathbf { j } - 8 \mathbf { k } ) + \lambda ( 3 \mathbf { i } + 3 \mathbf { j } - 4 \mathbf { k } ) \\ & l _ { 2 } : \mathbf { r } = ( 7 \mathbf { i } - 6 \mathbf { j } + 14 \mathbf { k } ) + \mu ( 2 \mathbf { i } - 3 \mathbf { j } + 2 \mathbf { k } ) \end{aligned}$$ where \(\lambda\) and \(\mu\) are scalar parameters.

1. Show that \(l _ { 1 }\) and \(l _ { 2 }\) meet and find the position vector of their point of intersection, \(B\).
2. Find the acute angle between the lines \(l _ { 1 }\) and \(l _ { 2 }\) The point \(A\) has position vector \(- 5 \mathbf { i } - 3 \mathbf { j } + 16 \mathbf { k }\)
3. Show that \(A\) lies on \(l _ { 1 }\) The point \(C\) lies on the line \(l _ { 1 }\) where \(\overrightarrow { A B } = \overrightarrow { B C }\)
4. Find the position vector of \(C\).
   
   \section*{"}

9 \end{array} \right)$$ where \(\mu\) is a scalar parameter.\\ (b) Show that \(l _ { 1 }\) and \(l _ { 2 }\) do not meet. The point \(C\) is on \(l _ { 2 }\) where \(\mu = - 1\)\\ (c) Find the acute angle between \(A C\) and \(l _ { 2 }\) Give your answer in degrees to one decimal place.\\

1. (a) Find the derivative with respect to \(y\) of

$$\frac { 1 } { ( 1 + 2 \ln y ) ^ { 2 } }$$ (b) Hence find a general solution to the differential equation $$3 \operatorname { cosec } ( 2 x ) \frac { \mathrm { d } y } { \mathrm {~d} x } = y ( 1 + 2 \ln y ) ^ { 3 } \quad y > 0 \quad - \frac { \pi } { 2 } < x < \frac { \pi } { 2 }$$ (c) Show that the particular solution of this differential equation for which \(y = 1\) at \(x = \frac { \pi } { 6 }\) is given by $$y = \mathrm { e } ^ { A \sec x - \frac { 1 } { 2 } }$$ where \(A\) is an irrational number to be found. \includegraphics[max width=\textwidth, alt={}, center]{fe07afad-9cfc-48c0-84f1-5717f81977d4-32_2649_1894_109_173}

6. The points \(A\) and \(B\) have position vectors \(2 \mathbf { i } + 6 \mathbf { j } - \mathbf { k }\) and \(3 \mathbf { i } + 4 \mathbf { j } + \mathbf { k }\) respectively. The line \(l _ { 1 }\) passes through the points \(A\) and \(B\).

1. Find the vector \(\overrightarrow { A B }\).
2. Find a vector equation for the line \(l _ { 1 }\). A second line \(l _ { 2 }\) passes through the origin and is parallel to the vector \(\mathbf { i } + \mathbf { k }\). The line \(l _ { 1 }\) meets the line \(l _ { 2 }\) at the point \(C\).
3. Find the acute angle between \(l _ { 1 }\) and \(l _ { 2 }\).
4. Find the position vector of the point \(C\).

15
3 \end{array} \right) + \mu \left( \begin{array} { r } 2
- 3
1 \end{array} \right)$$ where \(\lambda\) and \(\mu\) are scalar parameters.

1. Show that \(l _ { 1 }\) and \(l _ { 2 }\) meet and find the position vector of their point of intersection \(A\).
2. Find, to the nearest \(0.1 ^ { \circ }\), the acute angle between \(l _ { 1 }\) and \(l _ { 2 }\). The point \(B\) has position vector \(\left( \begin{array} { r } 5 \\ - 1 \\ 1 \end{array} \right)\).
3. Show that \(B\) lies on \(l _ { 1 }\).
4. Find the shortest distance from \(B\) to the line \(l _ { 2 }\), giving your answer to 3 significant figures.\\ 7. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{9d513d77-b8f9-4223-832f-f566c5f50457-10_643_999_276_475} \captionsetup{labelformat=empty} \caption{Figure 3}
   \end{figure} Figure 3 shows part of the curve \(C\) with parametric equations $$x = \tan \theta , \quad y = \sin \theta , \quad 0 \leqslant \theta < \frac { \pi } { 2 }$$ The point \(P\) lies on \(C\) and has coordinates \(\left( \sqrt { } 3 , \frac { 1 } { 2 } \sqrt { } 3 \right)\).
5. Find the value of \(\theta\) at the point \(P\). The line \(l\) is a normal to \(C\) at \(P\). The normal cuts the \(x\)-axis at the point \(Q\).
6. Show that \(Q\) has coordinates \(( k \sqrt { } 3,0 )\), giving the value of the constant \(k\). The finite shaded region \(S\) shown in Figure 3 is bounded by the curve \(C\), the line \(x = \sqrt { } 3\) and the \(x\)-axis. This shaded region is rotated through \(2 \pi\) radians about the \(x\)-axis to form a solid of revolution.
7. Find the volume of the solid of revolution, giving your answer in the form \(p \pi \sqrt { } 3 + q \pi ^ { 2 }\), where \(p\) and \(q\) are constants. 8. (a) Find \(\int ( 4 y + 3 ) ^ { - \frac { 1 } { 2 } } \mathrm {~d} y\)
8. Given that \(y = 1.5\) at \(x = - 2\), solve the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { \sqrt { } ( 4 y + 3 ) } { x ^ { 2 } }$$ giving your answer in the form \(y = \mathrm { f } ( x )\).

4. With respect to a fixed origin \(O\), the line \(l _ { 1 }\) has vector equation $$\mathbf { r } = \left( \begin{array} { r } - 9

5
- 3
p \end{array} \right) + \lambda \left( \begin{array} { r } 0
1
- 3 \end{array} \right) , \quad l _ { 2 } : \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.
3. Show that, if \(y = \operatorname { cosec } x\), then \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) can be expressed as \(- \operatorname { cosec } x \cot x\).
4. Solve the differential equation $$\frac { \mathrm { d } x } { \mathrm {~d} t } = - \sin x \tan x \cot t$$ given that \(x = \frac { 1 } { 6 } \pi\) when \(t = \frac { 1 } { 2 } \pi\). 8
5. Given that \(\frac { 2 t } { ( t + 1 ) ^ { 2 } }\) can be expressed in the form \(\frac { A } { t + 1 } + \frac { B } { ( t + 1 ) ^ { 2 } }\), find the values of the constants \(A\) and \(B\).
6. Show that the substitution \(t = \sqrt { 2 x - 1 }\) transforms \(\int \frac { 1 } { x + \sqrt { 2 x - 1 } } \mathrm {~d} x\) to \(\int \frac { 2 t } { ( t + 1 ) ^ { 2 } } \mathrm {~d} t\).
7. Hence find the exact value of \(\int _ { 1 } ^ { 5 } \frac { 1 } { x + \sqrt { 2 x - 1 } } \mathrm {~d} x\). 9 The parametric equations of a curve are $$x = 2 \theta + \sin 2 \theta , \quad y = 4 \sin \theta$$ and part of its graph is shown below. \includegraphics[max width=\textwidth, alt={}, center]{b8ba126f-c5fa-4828-9439-e5162a03ca5b-3_646_1150_1050_500}
8. Find the value of \(\theta\) at \(A\) and the value of \(\theta\) at \(B\).
9. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \sec \theta\).
10. At the point \(C\) on the curve, the gradient is 2 . Find the coordinates of \(C\), giving your answer in an exact form.

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.\\ 8. A small town had a population of 9000 in the year 2001. In a model, it is assumed that the population of the town, \(P\), at time \(t\) years after 2001 satisfies the differential equation $$\frac { \mathrm { d } P } { \mathrm {~d} t } = 0.05 P \mathrm { e } ^ { - 0.05 t }$$
4. Show that, according to the model, the population of the town in 2011 will be 13300 to 3 significant figures.
5. Find the value which the population of the town will approach in the long term, according to the model.\\ 9. A curve has parametric equations $$x = t ( t - 1 ) , \quad y = \frac { 4 t } { 1 - t } , \quad t \neq 1$$
6. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(t\). The point \(P\) on the curve has parameter \(t = - 1\).
7. Show that the tangent to the curve at \(P\) has the equation $$x + 3 y + 4 = 0$$ The tangent to the curve at \(P\) meets the curve again at the point \(Q\).
8. Find the coordinates of \(Q\).

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

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