4.04c Scalar product: calculate and use for angles

9

1. Find the value of \(k\) such that \(\left( \begin{array} { l } 1 \\ 2 \\ 1 \end{array} \right)\) and \(\left( \begin{array} { r } - 2 \\ 3 \\ k \end{array} \right)\) are perpendicular. Two lines have equations \(l _ { 1 } : \mathbf { r } = \left( \begin{array} { l } 3 \\ 2 \\ 7 \end{array} \right) + \lambda \left( \begin{array} { r } 1 \\ - 1 \\ 3 \end{array} \right)\) and \(l _ { 2 } : \mathbf { r } = \left( \begin{array} { l } 6 \\ 5 \\ 2 \end{array} \right) + \mu \left( \begin{array} { r } 2 \\ 1 \\ - 1 \end{array} \right)\).
2. Find the point of intersection of \(l _ { 1 }\) and \(l _ { 2 }\).
3. The vector \(\left( \begin{array} { l } 1 \\ a \\ b \end{array} \right)\) is perpendicular to the lines \(l _ { 1 }\) and \(l _ { 2 }\). Find the values of \(a\) and \(b\). \section*{END OF QUESTION PAPER} \section*{Copyright Information:} }{www.ocr.org.uk}) after the live examination series.
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8 The equation of a plane is \(4 x + 2 y + z = 7\).
The point \(A\) has coordinates \(( 9,6,1 )\) and the point \(B\) is the reflection of \(A\) in the plane.
Find the coordinates of the point \(B\).

4 Determine the acute angle between the line \(\mathbf { r } = \left( \begin{array} { c } - \sqrt { 3 } \\ 1 \\ 3 \end{array} \right) + \lambda \left( \begin{array} { c } 1 \\ 2 \sqrt { 3 } \\ - \sqrt { 3 } \end{array} \right)\) and the \(y\)-axis.

8 The points \(P , Q\) and \(R\) have coordinates \(( 0,2,3 ) , ( 2,0,1 )\) and \(( 1,3,0 )\) respectively.
The acute angle between the line segments \(P Q\) and \(P R\) is \(\theta\).

1. Show that \(\sin \theta = \frac { 2 } { 11 } \sqrt { 22 }\). The triangle \(P Q R\) lies in the plane \(\Pi\).
2. Determine an equation for \(\Pi\), giving your answer in the form \(\mathrm { ax } + \mathrm { by } + \mathrm { cz } = \mathrm { d }\), where \(a , b , c\) and \(d\) are integers. The point \(S\) has coordinates \(( 5,3 , - 1 )\).
3. By finding the shortest distance between \(S\) and the plane \(\Pi\), show that the volume of the tetrahedron \(P Q R S\) is \(\frac { 14 } { 3 }\).
   [0pt] [The volume of a tetrahedron is \(\frac { 1 } { 3 } \times\) area of base × perpendicular height] The tetrahedron \(P Q R S\) is transformed to the tetrahedron \(\mathrm { P } ^ { \prime } \mathrm { Q } ^ { \prime } \mathrm { R } ^ { \prime } \mathrm { S } ^ { \prime }\) by a rotation about the \(y\)-axis.
   The \(x\)-coordinate of \(S ^ { \prime }\) is \(2 \sqrt { 2 }\).
4. By using the matrix for a rotation by angle \(\theta\) about the \(y\)-axis, as given in the Formulae Booklet, determine in exact form the possible coordinates of \(R ^ { \prime }\).

3 The line \(l _ { 1 }\) has equation \(\mathbf { r } = \left( \begin{array} { r } 1 \\ - 3 \\ 3 \end{array} \right) + \lambda \left( \begin{array} { r } 3 \\ 2 \\ - 2 \end{array} \right)\).
The plane \(\Pi\) has equation \(\mathbf { r } \cdot \left( \begin{array} { r } 2 \\ - 5 \\ - 3 \end{array} \right) = 4\).

1. Find the position vector of the point of intersection of \(l _ { 1 }\) and \(\Pi\).
2. Find the acute angle between \(l _ { 1 }\) and \(\Pi\). \(A\) is the point on \(l _ { 1 }\) where \(\lambda = 1\). \(l _ { 2 }\) is the line with the following properties.

8 One end of a light elastic string of natural length \(l \mathrm {~m}\) and modulus of elasticity \(\lambda \mathrm { N }\) is attached to a particle \(A\) of mass \(m \mathrm {~kg}\). The other end of the string is attached to a fixed point \(O\) which is on a horizontal surface. The surface is modelled as being smooth and \(A\) moves in a circular path around \(O\) with constant speed \(v \mathrm {~ms} ^ { - 1 }\). The extension of the string is denoted by \(x \mathrm {~m}\).

1. Show that \(x\) satisfies \(\lambda x ^ { 2 } + \lambda | x - | m v ^ { 2 } = 0\).
2. By solving the equation in part (a) and using a binomial series, show that if \(\lambda\) is very large then \(\lambda \mathrm { x } \approx \mathrm { mv } ^ { 2 }\).
3. By considering the tension in the string, explain how the result obtained when \(\lambda\) is very large relates to the situation when the string is inextensible. The nature of the horizontal surface is such that the modelling assumption that it is smooth is justifiable provided that the speed of the particle does not exceed \(7 \mathrm {~ms} ^ { - 1 }\). In the case where \(m = 0.16\) and \(\lambda = 260\), the extension of the string is measured as being 3.0 cm .
4. Estimate the value of \(v\).
5. Explain whether the value of \(v\) means that the modelling assumption is necessarily justifiable in this situation.

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

1. 1. Find the vector \(\overrightarrow { A B }\).
   2. Calculate the acute angle between \(\overrightarrow { A B }\) and the line \(l _ { 1 }\), giving your answer to the nearest \(0.1 ^ { \circ }\).
2. The point \(D\) lies on \(l _ { 1 }\) where \(\lambda = 2\). The line \(l _ { 2 }\) passes through \(D\) and is parallel to \(A B\).
   1. Find a vector equation of line \(l _ { 2 }\) with parameter \(\mu\).
   2. The diagram shows a symmetrical trapezium \(A B C D\), with angle \(D A B\) equal to angle \(A B C\). \includegraphics[max width=\textwidth, alt={}, center]{5fe2527a-33da-4076-b3fa-4cab545336ec-9_620_675_1197_726} The point \(C\) lies on line \(l _ { 2 }\). The length of \(A D\) is equal to the length of \(B C\). Find the coordinates of \(C\).

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

1. 1. Find the vector \(\overrightarrow { A B }\).
   2. Find the acute angle between \(A B\) and the line \(l\), giving your answer to the nearest degree.
2. The point \(C\) lies on the line \(l\) such that the angle \(A B C\) is a right angle. Given that \(A B C D\) is a rectangle, find the coordinates of the point \(D\).

6

1. The points \(A , B\) and \(C\) have coordinates \(( 3,1 , - 6 ) , ( 5 , - 2,0 )\) and \(( 8 , - 4 , - 6 )\) respectively.
   1. Show that the vector \(\overrightarrow { A C }\) is given by \(\overrightarrow { A C } = n \left[ \begin{array} { r } 1 \\ - 1 \\ 0 \end{array} \right]\), where \(n\) is an integer.
   2. Show that the acute angle \(A C B\) is given by \(\cos ^ { - 1 } \left( \frac { 5 \sqrt { 2 } } { 14 } \right)\).
2. Find a vector equation of the line \(A C\).
3. The point \(D\) has coordinates \(( 6 , - 1 , p )\). It is given that the lines \(A C\) and \(B D\) intersect.
   1. Find the value of \(p\).
   2. Show that \(A B C D\) is a rhombus, and state the length of each of its sides.

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 \(C\).

\(\mathbf { 7 } \quad\) The line \(l _ { 1 }\) has equation \(\mathbf { r } = \left[ \begin{array} { r } 0 \\ - 2 \\ q \end{array} \right] + \lambda \left[ \begin{array} { r } 2 \\ 0 \\ - 1 \end{array} \right]\), where \(q\) is an integer. The line \(l _ { 2 }\) has equation \(\mathbf { r } = \left[ \begin{array} { l } 8 \\ 3 \\ 5 \end{array} \right] + \mu \left[ \begin{array} { l } 2 \\ 5 \\ 4 \end{array} \right]\). The lines \(l _ { 1 }\) and \(l _ { 2 }\) intersect at the point \(P\).

1. Show that \(q = 4\) and find the coordinates of \(P\).
2. Show that \(l _ { 1 }\) and \(l _ { 2 }\) are perpendicular.
3. The point \(A\) lies on the line \(l _ { 1 }\) where \(\lambda = 1\).
   1. Find \(A P ^ { 2 }\).
   2. The point \(B\) lies on the line \(l _ { 2 }\) so that the right-angled triangle \(A P B\) is isosceles. Find the coordinates of the two possible positions of \(B\).

9. The equations of the lines \(l _ { 1 }\) and \(l _ { 2 }\) are given by $$\begin{array} { l l } l _ { 1 } : & \mathbf { r } = \mathbf { i } + 3 \mathbf { j } + 5 \mathbf { k } + \lambda ( \mathbf { i } + 2 \mathbf { j } - \mathbf { k } ) , \\ l _ { 2 } : & \mathbf { r } = - 2 \mathbf { i } + 3 \mathbf { j } - 4 \mathbf { k } + \mu ( 2 \mathbf { i } + \mathbf { j } + 4 \mathbf { k } ) , \end{array}$$ where \(\lambda\) and \(\mu\) are parameters.

1. Show that \(l _ { 1 }\) and \(l _ { 2 }\) intersect and find the coordinates of \(Q\), their point of intersection.
2. Show that \(l _ { 1 }\) is perpendicular to \(l _ { 2 }\). The point \(P\) with \(x\)-coordinate 3 lies on the line \(l _ { 1 }\) and the point \(R\) with \(x\)-coordinate 4 lies on the line \(l _ { 2 }\).
3. Find, in its simplest form, the exact area of the triangle \(P Q R\). END

4. Relative to a fixed origin \(O\), the point \(A\) has position vector \(4 \mathbf { i } + 8 \mathbf { j } - \mathbf { k }\), and the point \(B\) has position vector \(7 \mathbf { i } + 14 \mathbf { j } + 5 \mathbf { k }\).

1. Find the vector \(\overrightarrow { A B }\).
2. Calculate the cosine of \(\angle O A B\).
3. Show that, for all values of \(\lambda\), the point P with position vector \(\lambda \mathbf { i } + 2 \lambda \mathbf { j } + ( 2 \lambda - 9 ) \mathbf { k }\) lies on the line through \(A\) and \(B\).
4. Find the value of \(\lambda\) for which \(O P\) is perpendicular to \(A B\).
5. Hence find the coordinates of the foot of the perpendicular from \(O\) to \(A B\).

3. Relative to a fixed origin, \(O\), the line \(l\) has the equation $$\mathbf { r } = ( \mathbf { i } + p \mathbf { j } - 5 \mathbf { k } ) + \lambda ( 3 \mathbf { i } - \mathbf { j } + q \mathbf { k } ) ,$$ where \(p\) and \(q\) are constants and \(\lambda\) is a scalar parameter.
Given that the point \(A\) with coordinates \(( - 5,9 , - 9 )\) lies on \(l\),

1. find the values of \(p\) and \(q\),
2. show that the point \(B\) with coordinates \(( 25 , - 1,11 )\) also lies on \(l\). The point \(C\) lies on \(l\) and is such that \(O C\) is perpendicular to \(l\).
3. Find the coordinates of \(C\).
4. Find the ratio \(A C : C B\) 3. continued

5. Relative to a fixed origin, the points \(A , B\) and \(C\) have position vectors ( \(2 \mathbf { i } - \mathbf { j } + 6 \mathbf { k }\) ), \(( 5 \mathbf { i } - 4 \mathbf { j } )\) and \(( 7 \mathbf { i } - 6 \mathbf { j } - 4 \mathbf { k } )\) respectively.

1. Show that \(A , B\) and \(C\) all lie on a single straight line.
2. Write down the ratio \(A B : B C\) The point \(D\) has position vector \(( 3 \mathbf { i } + \mathbf { j } + 4 \mathbf { k } )\).
3. Show that \(A D\) is perpendicular to \(B D\).
4. Find the exact area of triangle \(A B D\).
   5. continued

10 A vector \(\mathbf { v }\) has magnitude 1 unit. The angle between \(\mathbf { v }\) and the positive \(z\)-axis is \(60 ^ { \circ }\), and \(\mathbf { v }\) is parallel to the plane \(x - 2 y = 0\). Given that \(\mathbf { v } = a \mathbf { i } + b \mathbf { j } + c \mathbf { k }\), where \(a , b\) and \(c\) are all positive, find \(\mathbf { v }\). \section*{END OF QUESTION PAPER}

3 Three planes have the following equations. $$\begin{aligned} 2 x - 3 y + z & = - 3 \\ x - 4 y + 2 z & = 1 \\ - 3 x - 2 y + 3 z & = 14 \end{aligned}$$

1. 1. Write the system of equations in matrix form.
   2. Hence find the point of intersection of the planes.
2. In this question you must show detailed reasoning. Find the acute angle between the planes \(2 x - 3 y + z = - 3\) and \(x - 4 y + 2 z = 1\).

12 Three intersecting lines \(L _ { 1 } , L _ { 2 }\) and \(L _ { 3 }\) have equations \(L _ { 1 } : \frac { x } { 2 } = \frac { y } { 3 } = \frac { z } { 1 } , \quad L _ { 2 } : \frac { x } { 1 } = \frac { y } { 2 } = \frac { z } { - 4 } \quad\) and \(\quad L _ { 3 } : \frac { x - 1 } { 1 } = \frac { y - 2 } { 1 } = \frac { z + 4 } { 5 }\).
Find the area of the triangle enclosed by these lines.

13 The points A and B have coordinates \(( 4,0 , - 1 )\) and \(( 10,4 , - 3 )\) respectively. The planes \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\) have equations \(x - 2 y = 5\) and \(2 x + 3 y - z = - 4\) respectively.

1. Find the acute angle between the line AB and the plane \(\Pi _ { 1 }\).
2. Show that the line AB meets \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\) at the same point, whose coordinates should be specified.
3. 1. Find \(( \mathbf { i } - 2 \mathbf { j } ) \times ( 2 \mathbf { i } + 3 \mathbf { j } - \mathbf { k } )\).
   2. Hence find the acute angle between the planes \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\).
   3. Find the shortest distance between the point A and the line of intersection of the planes \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\).

5

1. Given that \(\mathbf { u } = \left( \begin{array} { r } - 2 \\ 1 \\ 2 \end{array} \right) , \mathbf { v } = \left( \begin{array} { l } a \\ 0 \\ 1 \end{array} \right)\) and \(\mathbf { u } \times \mathbf { v } = \left( \begin{array} { l } 1 \\ b \\ 3 \end{array} \right)\), find \(a\) and \(b\).
2. Using \(\mathbf { u } \times \mathbf { v }\), determine the angle between the vectors \(\mathbf { u }\) and \(\mathbf { v }\), given that this angle is acute.

8

1. Given that the lines \(\mathbf { r } = \left( \begin{array} { l } 0 \\ 2 \\ 2 \end{array} \right) + \lambda \left( \begin{array} { r } - 1 \\ 1 \\ 3 \end{array} \right)\) and \(\mathbf { r } = \left( \begin{array} { r } - 1 \\ 2 \\ k \end{array} \right) + \mu \left( \begin{array} { l } 2 \\ 3 \\ 4 \end{array} \right)\) meet, determine \(k\).
2. In this question you must show detailed reasoning. Find the acute angle between the two lines.

11

1. Given that \(\mathbf { u } = \lambda \mathbf { i } + \mathbf { j } - 3 \mathbf { k }\) and \(\mathbf { v } = \mathbf { i } + 2 \mathbf { j } - 2 \mathbf { k }\), find the following, giving your answers in terms of \(\lambda\).
   1. u.v
   2. \(\mathbf { u } \times \mathbf { v }\)
2. Hence determine
   1. the acute angle between the planes \(2 x + y - 3 z = 10\) and \(x + 2 y - 2 z = 10\),
   2. the shortest distance between the lines \(\frac { x - 3 } { 3 } = \frac { y } { 1 } = \frac { z - 2 } { - 3 }\) and \(\frac { x } { 1 } = \frac { y - 4 } { 2 } = \frac { z + 2 } { - 2 }\), giving your answer as a multiple of \(\sqrt { 2 }\).

2. The position vectors of the points \(A , B , C , D\) are given by \(\mathbf { a } = 2 \mathbf { i } + 3 \mathbf { j } - \mathbf { k }\), \(\mathbf { b } = 4 \mathbf { j } + 5 \mathbf { k }\), \(\mathbf { c } = 7 \mathbf { i } - 3 \mathbf { k }\), \(\mathbf { d } = - 3 \mathbf { i } - \mathbf { j } - 5 \mathbf { k }\),
respectively.

1. Find the vector equations of the lines \(A B\) and \(C D\).
2. Determine whether or not the lines \(A B\) and \(C D\) are perpendicular.

7. The vector equations of the lines \(L _ { 1 } , L _ { 2 } , L _ { 3 }\) are given by $$\begin{aligned} & \mathbf { r } = 3 \mathbf { i } + 2 \mathbf { j } + \mathbf { k } + \lambda ( 2 \mathbf { i } + n \mathbf { j } + \mathbf { k } ) \\ & \mathbf { r } = 5 \mathbf { i } - 3 \mathbf { j } - 4 \mathbf { k } + \mu ( 3 \mathbf { i } + \mathbf { j } - 3 \mathbf { k } ) \\ & \mathbf { r } = 6 \mathbf { i } - 3 \mathbf { j } + 2 \mathbf { k } + v ( p \mathbf { i } + 3 \mathbf { j } + 4 \mathbf { k } ) \end{aligned}$$ respectively, where \(n\) and \(p\) are constants.
The line \(L _ { 1 }\) is perpendicular to the line \(L _ { 2 }\). The line \(L _ { 1 }\) is also perpendicular to the line \(L _ { 3 }\).

1. Show that the value of \(n\) is - 3 and find the value of \(p\).
2. Find the acute angle between the lines \(L _ { 2 }\) and \(L _ { 3 }\).

9. Two planes, \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\), are defined by $$\begin{aligned} & \Pi _ { 1 } : 4 x - 3 y + 2 z = 5 \\ & \Pi _ { 2 } : 6 x + y + z = 9 \end{aligned}$$

1. Find the acute angle between the planes \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\). Give your answer correct to three significant figures.
2. Find the perpendicular distance from the point \(A ( 5 , - 2 , - 6 )\) to the plane \(\Pi _ { 1 }\).
3. 1. Show that the point \(B ( 5,5,0 )\) lies on \(\Pi _ { 1 }\) and that the point \(C ( 1,3,0 )\) lies on \(\Pi _ { 2 }\).
   2. State an equation of a plane that contains the points \(B\) and \(C\).
      Additional page, if required. Write the question number(s) in the left-hand margin. Additional page, if required. Write the question number(s) in the left-hand margin. \section*{PLEASE DO NOT WRITE ON THIS PAGE}