1.10b Vectors in 3D: i,j,k notation

13. Relative to a fixed origin \(O\)

• the point \(P\) has position vector \(( 0 , - 1,2 )\)
• the point \(Q\) has position vector \(( 1,1,5 )\)
• the point \(R\) has position vector ( \(3,5 , m\) )
  where \(m\) is a constant.
  Given that \(P , Q\) and \(R\) lie on a straight line,
  a. find the value of \(m\)

The line segment \(O Q\) is extended to a point \(T\) so that \(\overrightarrow { R T }\) is parallel to \(\overrightarrow { O P }\) b. Show that \(| \overrightarrow { O T } | = 9 \sqrt { 3 }\).

Relative to a fixed origin \(O\),
the point \(A\) has position vector \(\mathbf { i } + 7 \mathbf { j } - 2 \mathbf { k }\),
the point \(B\) has position vector \(4 \mathbf { i } + 3 \mathbf { j } + 3 \mathbf { k }\),
and the point \(C\) has position vector \(2 \mathbf { i } + 10 \mathbf { j } + 9 \mathbf { k }\).
Given that \(A B C D\) is a parallelogram,

1. find the position vector of point \(D\). The vector \(\overrightarrow { A X }\) has the same direction as \(\overrightarrow { A B }\).
   Given that \(| \overrightarrow { A X } | = 10 \sqrt { 2 }\),
2. find the position vector of \(X\).

Relative to a fixed origin \(O\),
the point \(A\) has position vector \(( 2 \mathbf { i } + 3 \mathbf { j } - 4 \mathbf { k } )\),
the point \(B\) has position vector ( \(4 \mathbf { i } - 2 \mathbf { j } + 3 \mathbf { k }\) ),
and the point \(C\) has position vector ( \(a \mathbf { i } + 5 \mathbf { j } - 2 \mathbf { k }\) ), where \(a\) is a constant and \(a < 0 D\) is the point such that \(\overrightarrow { A B } = \overrightarrow { B D }\).

1. Find the position vector of \(D\). Given \(| \overrightarrow { A C } | = 4\)
2. find the value of \(a\).

1. Relative to a fixed origin \(O\)

• the point \(A\) has position vector \(4 \mathbf { i } - 3 \mathbf { j } + 5 \mathbf { k }\)
• the point \(B\) has position vector \(4 \mathbf { j } + 6 \mathbf { k }\)
• the point \(C\) has position vector \(- 16 \mathbf { i } + p \mathbf { j } + 10 \mathbf { k }\) where \(p\) is a constant.
  Given that \(A , B\) and \(C\) lie on a straight line,
  1. find the value of \(p\).

The line segment \(O B\) is extended to a point \(D\) so that \(\overrightarrow { C D }\) is parallel to \(\overrightarrow { O A }\) (b) Find \(| \overrightarrow { O D } |\), writing your answer as a fully simplified surd.

1. Relative to a fixed origin \(O\),

• \(A\) is the point with position vector \(12 \mathbf { i }\)
• \(B\) is the point with position vector \(16 \mathbf { j }\)
• \(C\) is the point with position vector \(( 50 \mathbf { i } + 136 \mathbf { j } )\)
• \(D\) is the point with position vector \(( 22 \mathbf { i } + 24 \mathbf { j } )\)
  1. Show that \(A D\) is parallel to \(B C\).

Points \(A , B , C\) and \(D\) are used to model the vertices of a running track in the shape of a quadrilateral. Runners complete one lap by running along all four sides of the track.
The lengths of the sides are measured in metres. Given that a particular runner takes exactly 5 minutes to complete 2 laps,

calculate the average speed of this runner, giving the answer in kilometres per hour.

7. \begin{figure}[h] \end{figure} Figure 2 shows a sketch of the straight line \(l\).
Line \(l\) passes through the points \(A\) and \(B\).
Relative to a fixed origin \(O\)

• the point \(A\) has position vector \(2 \mathbf { i } - 3 \mathbf { j } + 5 \mathbf { k }\)
• the point \(B\) has position vector \(5 \mathbf { i } + 6 \mathbf { j } + 8 \mathbf { k }\)
  1. Find \(\overrightarrow { A B }\)

Given that a point \(P\) lies on \(l\) such that $$| \overrightarrow { A P } | = 2 | \overrightarrow { B P } |$$

find the possible position vectors of \(P\).

9 In this question the horizontal unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are in the directions east and north respectively.
A model ship of mass 2 kg is moving so that its acceleration vector \(\mathbf { a m s } ^ { - 2 }\) at time \(t\) seconds is given by \(\mathbf { a } = 3 ( 2 t - 5 ) \mathbf { i } + 4 \mathbf { j }\). When \(t = T\), the magnitude of the horizontal force acting on the ship is 10 N . Find the possible values of \(T\).

7 \includegraphics[max width=\textwidth, alt={}, center]{a1f4ccbd-f5ed-437a-ae76-c4925ce86e25-06_648_586_255_244} The diagram shows the parallelogram \(O A C B\) where \(\overrightarrow { O A } = 2 \mathbf { i } + 4 \mathbf { j }\) and \(\overrightarrow { O B } = 4 \mathbf { i } - 3 \mathbf { j }\).

1. Show that \(\cos A O B = - \frac { 2 \sqrt { 5 } } { 25 }\).
2. Hence find the exact value of \(\sin A O B\).
3. Determine the area of \(O A C B\).

9 Two forces \(( 3 \mathbf { i } + 2 \mathbf { j } ) \mathrm { N }\) and \(\mathbf { F N }\) act on a particle \(P\) of mass 4 kg .
Given that the acceleration of \(P\) is \(( - 2 \mathbf { i } + 3 \mathbf { j } ) \mathrm { ms } ^ { - 2 }\), calculate \(\mathbf { F }\).

5 In this question, the unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are horizontal and vertically upwards respectively. A particle has mass 2.5 kg .

1. Write the weight of the particle as a vector. The particle moves under the action of its weight and two external forces ( \(3 \mathbf { i } - 2 \mathbf { j }\) ) N and \(( - \mathbf { i } + 18 \mathbf { j } ) N\).
2. Find the acceleration of the particle, giving your answer in vector form.

7 In this question the unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are directed east and north respectively. A canal narrowboat of mass 9 tonnes is pulled by two ropes. The tensions in the ropes are \(( 450 \mathbf { i } + 20 \mathbf { j } ) \mathbf { N }\) and \(( 420 \mathbf { i } - 20 \mathbf { j } ) \mathbf { N }\). The boat experiences a resistance to motion \(\mathbf { R }\) of magnitude 300 N .

1. Explain what it means to model the boat as a particle. The boat is travelling in a straight line due east.
2. Find the equation of motion of the boat.
3. Find the acceleration of the boat giving your answer as a vector.

4 In this question, the \(x\) and \(y\) directions are horizontal and vertically upwards respectively.
A particle of mass 1.5 kg is in equilibrium under the action of its weight and forces \(\mathbf { F } _ { 1 } = \binom { 4 } { - 2 } \mathrm {~N}\) and \(\mathbf { F } _ { 2 }\). and \(\mathbf { F } _ { 2 }\).

1. Find the force \(\mathbf { F } _ { 2 }\). The force \(\mathbf { F } _ { 2 }\) is changed to \(\binom { 2 } { 20 } \mathrm {~N}\).
2. Find the acceleration of the particle.

3 Forces \(\mathbf { F } _ { 1 } = ( 2 \mathbf { i } + 9 \mathbf { j } ) \mathbf { N }\) and \(\mathbf { F } _ { 2 } = ( - \mathbf { i } + \mathbf { j } ) \mathbf { N }\) act on a particle. A third force \(\mathbf { F } _ { 3 }\) acts so that the particle is in equilibrium under the action of the three forces. Find the force \(\mathbf { F } _ { 3 }\).

15 Fig. 15 shows a particle of mass \(m \mathrm {~kg}\) on a smooth plane inclined at \(30 ^ { \circ }\) to the horizontal. Unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are parallel and perpendicular to the plane, in the directions shown. \begin{figure}[h] \end{figure}

1. Express the weight \(\mathbf { W }\) of the particle in terms of \(m , g , \mathbf { i }\) and \(\mathbf { j }\). The particle is held in equilibrium by a force \(\mathbf { F }\), and the normal reaction of the plane on the particle is denoted by \(\mathbf { R }\). The units for both \(\mathbf { F }\) and \(\mathbf { R }\) are newtons.
2. Write down an equation relating \(\mathbf { W } , \mathbf { R }\) and \(\mathbf { F }\).
3. Given that \(\mathbf { F } = 6 \mathbf { i } + 8 \mathbf { j }\),

10 Point A has position vector \(\left( \begin{array} { l } a \\ b \\ 0 \end{array} \right)\) where \(a\) and \(b\) can vary, point B has position vector \(\left( \begin{array} { l } 4 \\ 2 \\ 0 \end{array} \right)\) and point C has position vector \(\left( \begin{array} { l } 2 \\ 4 \\ 2 \end{array} \right)\). ABC is an isosceles triangle with \(\mathrm { AC } = \mathrm { AB }\).

1. Show that \(a - b + 1 = 0\).
2. Determine the position vector of A such that triangle ABC has minimum area. Answer all the questions.
   Section B (15 marks) The questions in this section refer to the article on the Insert. You should read the article before attempting the questions.

5 The tetrahedron \(T\), shown below, has vertices at \(O ( 0,0,0 ) , A ( 1,2,2 ) , B ( 2,1,2 )\) and \(C ( 2,2,1 )\). \includegraphics[max width=\textwidth, alt={}, center]{59fa1650-a296-471e-93b9-0988177cd89d-3_360_464_319_555} Diagram not drawn to scale Show that the surface area of \(T\) is \(\frac { 1 } { 2 } \sqrt { 3 } ( 1 + \sqrt { 51 } )\).

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 .

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 \(B\).