4.04g Vector product: a x b perpendicular vector

1 The points \(A , B\) and \(C\) have position vectors \(6 \mathbf { i } + 2 \mathbf { j } + 4 \mathbf { k } , 13 \mathbf { i } + 2 \mathbf { j } + 5 \mathbf { k }\) and \(16 \mathbf { i } + 6 \mathbf { j } + 3 \mathbf { k }\) respectively.

1. Using the vector product, calculate the area of triangle \(A B C\).
2. Hence find, in simplest surd form, the perpendicular distance from \(C\) to the line through \(A\) and \(B\).

1 The points \(A , B\) and \(C\) have position vectors \(\mathbf { a } = \left( \begin{array} { l } 1 \\ 1 \\ 3 \end{array} \right) , \mathbf { b } = \left( \begin{array} { r } 2 \\ 3 \\ - 1 \end{array} \right)\) and \(\mathbf { c } = \left( \begin{array} { r } - 5 \\ 1 \\ 2 \end{array} \right)\) respectively, relative to the origin \(O\).

1. Calculate, in its simplest exact form, the area of triangle \(O A B\).
2. Show that \(\mathbf { a } \times ( \mathbf { b } \times \mathbf { c } ) + \mathbf { b } \times ( \mathbf { c } \times \mathbf { a } ) + \mathbf { c } \times ( \mathbf { a } \times \mathbf { b } ) = \mathbf { 0 }\).

4 The equation of line \(l\) can be written in either of the following vector forms.

• \(\mathbf { r } = \mathbf { a } + \lambda \mathbf { b }\), where \(\lambda \in \mathbb { R }\)
• \(( \mathbf { r } - \mathbf { c } ) \times \mathbf { d } = \mathbf { 0 }\)
  1. Write down two equations involving the vectors \(\mathbf { a , b , c }\), and d, giving reasons for your answers.
  2. Determine the value of \(\mathbf { a } \cdot ( \mathbf { c } \times \mathbf { d } )\).

6 The points \(A\) and \(B\) have position vectors \(\mathbf { a } = \mathbf { i } + 2 \mathbf { j } + \mathbf { k }\) and \(\mathbf { b } = - 3 \mathbf { i } + 4 \mathbf { j } - 5 \mathbf { k }\) respectively.

1. Determine the area of triangle \(O A B\), giving your answer in an exact form. The point \(C\) lies on the line \(( \mathbf { r } - \mathbf { a } ) \times ( \mathbf { b } - \mathbf { a } ) = \mathbf { O }\) such that the area of triangle \(O A C\) is half the area of triangle \(O A B\).
2. Determine the two possible position vectors of \(C\).

2 The points \(A ( 1,2,2 ) , B ( 8,2,5 ) , C ( - 3,6,5 )\) and \(D ( - 10,6,2 )\) are the vertices of parallelogram \(A B C D\). Determine the area of \(A B C D\).

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 \(R ^ { \prime }\).

1

1. Find a vector which is perpendicular to both \(3 \mathbf { i } - 5 \mathbf { j } - \mathbf { k }\) and \(\mathbf { i } + 3 \mathbf { j } - 4 \mathbf { k }\). The equations of two lines are \(\mathbf { r } = 2 \mathbf { i } + 3 \mathbf { j } + 3 \mathbf { k } + \lambda ( \mathbf { i } - 2 \mathbf { j } + \mathbf { k } )\) and \(\mathbf { r } = \mathbf { i } + 11 \mathbf { j } - 4 \mathbf { k } + \mu ( - \mathbf { i } + 3 \mathbf { j } - 2 \mathbf { k } )\).
2. Show that the lines intersect, stating the point of intersection.

3 The points \(A\) and \(B\) have position vectors \(\mathbf { a } = \mathbf { i } + \mathrm { pj } + \mathrm { q } \mathbf { k }\) and \(\mathbf { b } = 2 \mathbf { i } + 3 \mathbf { j } + 2 \mathbf { k }\) respectively, relative to the origin \(O\).

1. Determine the value of \(p\) and the value of \(q\) for which \(\mathbf { a } \times \mathbf { b } = 2 \mathbf { i } + 6 \mathbf { j } - 1 \mathbf { 1 } \mathbf { k }\).
2. The point \(C\) has coordinates ( \(d , e , f\) ) and the tetrahedron \(O A B C\) has volume 7.
   1. Using the values of \(p\) and \(q\) found in part (a), find the possible relationships between \(d , e\) and \(f\).
   2. Explain the geometrical significance of these relationships.

4 The vectors \(\mathbf { a }\) and \(\mathbf { b }\) are given by \(\mathbf { a } = \left( \begin{array} { c } p - 1 \\ q + 2 \\ 2 r - 3 \end{array} \right)\) and \(\mathbf { b } = \left( \begin{array} { c } 2 p + 4 \\ 2 q - 5 \\ r + 3 \end{array} \right)\), where \(p , q\) and \(r\) are real numbers.

1. Given that \(\mathbf { b }\) is not a multiple of \(\mathbf { a }\) and that \(\mathbf { a } \times \mathbf { b } = \mathbf { 0 }\), determine all possible sets of values of \(p , q\) and \(r\).
2. You are given instead that \(\mathbf { b } = \lambda \mathbf { a }\), where \(\lambda\) is an integer with \(| \lambda | > 1\). By writing each of \(p , q\) and \(r\) in terms of \(\lambda\), show that there is a unique value of \(\lambda\) for which \(p , q\) and \(r\) are all integers, stating this set of values of \(p , q\) and \(r\).

4 Points \(A , B\) and \(C\) have position vectors \(\mathbf { a } , \mathbf { b }\) and \(\mathbf { c }\) respectively, relative to origin \(O\). It is given that \(\mathbf { b } \times \mathbf { c } = \mathbf { a }\) and that \(| \mathbf { a } | = 3\).

1. Determine each of the following as either a single vector or a scalar quantity.
   1. \(\mathbf { c } \times \mathbf { b }\)
   2. \(\mathbf { a } \times ( \mathbf { b } \times \mathbf { c } )\)
   3. \(\mathbf { a } \cdot ( \mathbf { b } \times \mathbf { c } )\)
2. Describe a geometrical relationship between the points \(O , A , B\) and \(C\) which can be deduced from
   1. the statement \(\mathbf { b } \times \mathbf { c } = \mathbf { a }\),
   2. the result of (a)(iii).

3 The points \(P , Q\) and \(R\) have position vectors \(\mathbf { p } = 2 \mathbf { i } + \mathbf { j } + 5 \mathbf { k } , \mathbf { q } = \mathbf { i } - \mathbf { j } + \mathbf { k }\) and \(\mathbf { r } = 2 \mathbf { i } + \mathbf { j } + t \mathbf { k }\) respectively, relative to the origin \(O\). Determine the value(s) of \(t\) in each of the following cases.

1. The line \(O R\) is parallel to \(\mathbf { p } \times \mathbf { q }\).
2. The volume of tetrahedron \(O P Q R\) is 13 .

2 Find the volume of tetrahedron OABC , where O is the origin, \(\mathrm { A } = ( 2,3,1 ) , \mathrm { B } = ( - 4,2,5 )\) and \(\mathrm { C } = ( 1,4,4 )\).

3. A system of forces acting on a rigid body consists of two forces \(\mathbf { F } _ { 1 }\) and \(\mathbf { F } _ { 2 }\) acting at a point \(A\) of the body, together with a couple of moment \(\mathbf { G } . \mathbf { F } _ { 1 } = ( \mathbf { i } + 2 \mathbf { j } - \mathbf { k } ) \mathrm { N }\) and \(\mathbf { F } _ { 2 } = ( - 2 \mathbf { i } + \mathbf { j } + 3 \mathbf { k } ) N\). The position vector of the point \(A\) is \(( \mathbf { i } + \mathbf { j } + \mathbf { k } ) \mathrm { m }\) and \(\mathbf { G } = ( 7 \mathbf { i } - 3 \mathbf { j } + 8 \mathbf { k } ) \mathrm { Nm }\). Given that the system is equivalent to a single force \(\mathbf { R }\),

1. find \(\mathbf { R }\),
2. find a vector equation for the line of action of \(\mathbf { R }\).
   (Total 9 marks) \section*{4.} \section*{Figure 1}
   \includegraphics[max width=\textwidth, alt={}]{43ce237f-c8e4-428a-b8cd-04673e62abb9-3_896_515_276_772}
   A thin uniform rod \(P Q\) has mass \(m\) and length \(3 a\). A thin uniform circular disc, of mass \(m\) and radius \(a\), is attached to the rod at \(Q\) in such a way that the rod and the diameter \(Q R\) of the disc are in a straight line with \(P R = 5 a\). The rod together with the disc form a composite body, as shown in Figure 1. The body is free to rotate about a fixed smooth horizontal axis \(L\) through \(P\), perpendicular to \(P Q\) and in the plane of the disc.

5. Two forces \(\mathbf { F } _ { 1 }\) and \(\mathbf { F } _ { 2 }\) act on a rigid body, where \(\mathbf { F } _ { 1 } = ( 3 \mathbf { i } + 4 \mathbf { j } - 6 \mathbf { k } ) \mathrm { N }\) and \(\mathbf { F } _ { 2 } = ( 5 \mathbf { i } - \mathbf { j } + 2 \mathbf { k } ) \mathrm { N }\).
The force \(\mathbf { F } _ { 1 }\) acts at the point with position vector \(( \mathbf { i } - 2 \mathbf { j } ) \mathrm { m }\), and the force \(\mathbf { F } _ { 2 }\) acts at the point with position vector ( \(3 \mathbf { i } - \mathbf { k }\) ) m. The two forces are equivalent to a single force \(\mathbf { F }\) acting at the point with position vector \(( \mathbf { i } - \mathbf { k } ) \mathrm { m }\), together with a couple \(\mathbf { G }\).

1. Find \(\mathbf { F }\).
2. Find the magnitude of \(\mathbf { G }\).
   (8)

3. A system of forces consists of two forces \(\mathbf { F } _ { 1 }\) and \(\mathbf { F } _ { 2 }\) acting on a rigid body. \(\mathbf { F } _ { 1 } = ( - 2 \mathbf { i } + \mathbf { j } - \mathbf { k } ) \mathrm { N }\) and acts at the point with position vector \(\mathbf { r } _ { 1 } = ( \mathbf { i } - \mathbf { j } + \mathbf { k } ) \mathrm { m }\). \(\mathbf { F } _ { 2 } = ( 3 \mathbf { i } - \mathbf { j } + 2 \mathbf { k } ) \mathrm { N }\) and acts at the point with position vector \(\mathbf { r } _ { 2 } = ( 4 \mathbf { i } - \mathbf { j } - 2 \mathbf { k } ) \mathrm { m }\).
Given that the system is equivalent to a single force \(\mathbf { R } \mathrm { N }\), acting at the point with position vector \(( 5 \mathbf { i } + \mathbf { j } - \mathbf { k } ) \mathrm { m }\), together with a couple \(\mathbf { G N m }\), find

1. \(\mathbf { R }\),
2. the magnitude of \(\mathbf { G }\).
   (9)

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.

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 \(\sqrt { 2 }\).

1. The surface of a horizontal tennis court is modelled as part of a horizontal plane, with the origin on the ground at the centre of the court, and

• i and j are unit vectors directed across the width and length of the court respectively
• \(\quad \mathbf { k }\) is a unit vector directed vertically upwards
• units are metres

After being hit, a tennis ball, modelled as a particle, moves along the path with equation $$\mathbf { r } = \left( - 4.1 + 9 \lambda - 2.3 \lambda ^ { 2 } \right) \mathbf { i } + ( - 10.25 + 15 \lambda ) \mathbf { j } + \left( 0.84 + 0.8 \lambda - \lambda ^ { 2 } \right) \mathbf { k }$$ where \(\lambda\) is a scalar parameter with \(\lambda \geqslant 0\) Assuming that the tennis ball continues on this path until it hits the ground,

1. find the value of \(\lambda\) at the point where the ball hits the ground. The direction in which the tennis ball is moving at a general point on its path is given by $$( 9 - 4.6 \lambda ) \mathbf { i } + 15 \mathbf { j } + ( 0.8 - 2 \lambda ) \mathbf { k }$$
2. Write down the direction in which the tennis ball is moving as it hits the ground.
3. Hence find the acute angle at which the tennis ball hits the ground, giving your answer in degrees to one decimal place. The net of the tennis court lies in the plane \(\mathbf { r } . \mathbf { j } = 0\)
4. Find the position of the tennis ball at the point where it is in the same plane as the net. The maximum height above the court of the top of the net is 0.9 m .
   Modelling the top of the net as a horizontal straight line,
5. state whether the tennis ball will pass over the net according to the model, giving a reason for your answer. With reference to the model,
6. decide whether the tennis ball will actually pass over the net, giving a reason for your answer.

4. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of a solid doorstop made of wood. The doorstop is modelled as a tetrahedron. Relative to a fixed origin \(O\), the vertices of the tetrahedron are \(A ( 2,1,4 )\), \(B ( 6,1,2 ) , C ( 4,10,3 )\) and \(D ( 5,8 , d )\), where \(d\) is a positive constant and the units are in centimetres.

1. Find the area of the triangle \(A B C\). Given that the volume of the doorstop is \(21 \mathrm {~cm} ^ { 3 }\)
2. find the value of the constant \(d\).

5. \begin{figure}[h] \end{figure} Figure 3 shows a solid display stand with parallel triangular faces \(A B C\) and \(D E F\). Triangle \(D E F\) is similar to triangle \(A B C\). With respect to a fixed origin \(O\), the points \(A , B\) and \(C\) have coordinates ( \(3 , - 3,1\) ), ( \(- 5,3,3\) ) and ( \(1,7,5\) ) respectively and the points \(D , E\) and \(F\) have coordinates ( \(2 , - 1,8\) ), ( \(- 2,2,9\) ) and ( \(1,4,10\) ) respectively. The units are in centimetres.

1. Show that the area of the triangular face \(D E F\) is \(\frac { 1 } { 2 } \sqrt { 339 } \mathrm {~cm} ^ { 2 }\)
2. Find, in \(\mathrm { cm } ^ { 3 }\), the exact volume of the display stand.

1. With respect to a fixed origin \(O\), the points \(A\), \(B\) and \(C\) have position vectors given by

$$\overrightarrow { O A } = 18 \mathbf { i } - 14 \mathbf { j } - 2 \mathbf { k } \quad \overrightarrow { O B } = - 7 \mathbf { i } - 5 \mathbf { j } + 3 \mathbf { k } \quad \overrightarrow { O C } = - 2 \mathbf { i } - 9 \mathbf { j } - 6 \mathbf { k }$$ The points \(O , A , B\) and \(C\) form the vertices of a tetrahedron.

1. Show that the area of the triangular face \(A B C\) of the tetrahedron is 108 to 3 significant figures.
2. Find the volume of the tetrahedron. An oak wood block is made in the shape of the tetrahedron, with centimetres taken for the units. The density of oak is \(0.85 \mathrm {~g} \mathrm {~cm} ^ { - 3 }\)
3. Determine the mass of the block, giving your answer in kg.

5. \begin{figure}[h] \end{figure} The points \(A ( 3,2 , - 4 ) , B ( 9 , - 4,2 ) , C ( - 6 , - 10,8 )\) and \(D ( - 4 , - 5,10 )\) are the vertices of a tetrahedron. The plane with equation \(z = 0\) cuts the tetrahedron into two pieces, one on each side of the plane. The edges \(A B , A C\) and \(A D\) of the tetrahedron intersect the plane at the points \(M , N\) and \(P\) respectively, as shown in Figure 1. Determine

1. the coordinates of the points \(M , N\) and \(P\),
2. the area of triangle \(M N P\),
3. the exact volume of the solid \(B C D P N M\).

1. The points \(A , B\) and \(C\) are the vertices of a triangle.

Given that

• \(\overrightarrow { A B } = \left( \begin{array} { l } p \\ 4 \\ 6 \end{array} \right)\) and \(\overrightarrow { A C } = \left( \begin{array} { l } q \\ 4 \\ 5 \end{array} \right)\) where \(p\) and \(q\) are constants
• \(\overrightarrow { A B } \times \overrightarrow { A C }\) is parallel to \(2 \mathbf { i } + 3 \mathbf { j } + 4 \mathbf { k }\)
  1. determine the value of \(p\) and the value of \(q\)
  2. Hence, determine the exact area of triangle \(A B C\)