1.10a Vectors in 2D: i,j notation and column vectors

6. The point \(A\) on a horizontal playground has position vector \(( 3 \mathbf { i } - 2 \mathbf { j } ) \mathrm { m }\). At time \(t = 0\), a girl kicks a ball from \(A\). The ball moves horizontally along the playground with constant velocity \(( 4 \mathbf { i } + 5 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\). Modelling the ball as a particle, find

1. the speed of the ball,
2. the position vector of the ball at time \(t\) seconds. The point \(B\) on the playground has position vector \(( \mathbf { i } + 6 \mathbf { j } ) \mathrm { m }\). At time \(t = T\) seconds, the ball is due east of \(B\).
3. Find the value of \(T\). A boy is running due east with constant speed \(\nu \mathrm { ms } ^ { - 1 }\). At the instant when the girl kicks the ball from \(A\), the boy is at \(B\). Given that the boy intercepts the ball,
4. find the value of \(v\). \includegraphics[max width=\textwidth, alt={}, center]{5f2d38d9-b719-4205-8cb0-caa959afc46f-23_68_47_2617_1886}

6. Two forces, \(( 4 \mathbf { i } - 5 \mathbf { j } ) \mathrm { N }\) and \(( p \mathbf { i } + q \mathbf { j } ) \mathrm { N }\), act on a particle \(P\) of mass \(m \mathrm {~kg}\). The resultant of the two forces is \(\mathbf { R }\). Given that \(\mathbf { R }\) acts in a direction which is parallel to the vector ( \(\mathbf { i } - 2 \mathbf { j }\) ),

1. find the angle between \(\mathbf { R }\) and the vector \(\mathbf { j }\),
2. show that \(2 p + q + 3 = 0\). Given also that \(q = 1\) and that \(P\) moves with an acceleration of magnitude \(8 \sqrt { } 5 \mathrm {~m} \mathrm {~s} ^ { - 2 }\), (c) find the value of \(m\).

7 The position vectors of the points \(A , B , C , D , G\) are given by $$\mathbf { a } = 6 \mathbf { i } + 4 \mathbf { j } + 8 \mathbf { k } , \quad \mathbf { b } = 2 \mathbf { i } + \mathbf { j } + 3 \mathbf { k } , \quad \mathbf { c } = \mathbf { i } + 5 \mathbf { j } + 4 \mathbf { k } , \quad \mathbf { d } = 3 \mathbf { i } + 6 \mathbf { j } + 5 \mathbf { k } , \quad \mathbf { g } = 3 \mathbf { i } + 4 \mathbf { j } + 5 \mathbf { k }$$ respectively.

1. The line through \(A\) and \(G\) meets the plane \(B C D\) at \(M\). Write down the vector equation of the line through \(A\) and \(G\) and hence show that the position vector of \(M\) is \(2 \mathbf { i } + 4 \mathbf { j } + 4 \mathbf { k }\).
2. Find the value of the ratio \(A G : A M\).
3. Find the position vector of the point \(P\) on the line through \(C\) and \(G\), such that \(\overrightarrow { C P } = \frac { 4 } { 3 } \overrightarrow { C G }\).
4. Verify that \(P\) lies in the plane \(A B D\).

5 The vectors \(\mathbf { p }\) and \(\mathbf { q }\) are given by $$\mathbf { p } = 8 \mathbf { i } + \mathbf { j } \text { and } \mathbf { q } = 4 \mathbf { i } - 7 \mathbf { j } .$$

1. Show that \(\mathbf { p }\) and \(\mathbf { q }\) are equal in magnitude.
2. Show that \(\mathbf { p } + \mathbf { q }\) is parallel to \(2 \mathbf { i } - \mathbf { j }\).
3. Draw \(\mathbf { p } + \mathbf { q }\) and \(\mathbf { p } - \mathbf { q }\) on the grid. Write down the angle between these two vectors.

3 A particle rests on a smooth, horizontal plane. Horizontal unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) lie in this plane. The particle is in equilibrium under the action of the three forces \(( - 3 \mathbf { i } + 4 \mathbf { j } ) \mathrm { N }\) and \(( 21 \mathbf { i } - 7 \mathbf { j } ) \mathrm { N }\) and \(\mathbf { R N }\).

1. Write down an expression for \(\mathbf { R }\) in terms of \(\mathbf { i }\) and \(\mathbf { j }\).
2. Find the magnitude of \(\mathbf { R }\) and the angle between \(\mathbf { R }\) and the \(\mathbf { i }\) direction.

2 A particle has a position vector \(\mathbf { r }\), where \(\mathbf { r } = 4 \mathbf { i } - 5 \mathbf { j }\) and \(\mathbf { i }\) and \(\mathbf { j }\) are unit vectors in the directions east and north respectively.

1. Sketch \(\mathbf { r }\) on a diagram showing \(\mathbf { i }\) and \(\mathbf { j }\) and the origin O .
2. Calculate the magnitude of \(\mathbf { r }\) and its direction as a bearing.
3. Write down the vector that has the same direction as \(\mathbf { r }\) and three times its magnitude.

2 A small box has weight \(\mathbf { W } \mathrm { N }\) and is held in equilibrium by two strings with tensions \(\mathbf { T } _ { 1 } \mathrm {~N}\) and \(\mathbf { T } _ { 2 } \mathrm {~N}\). This situation is shown in Fig. 2 which also shows the standard unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) that are horizontal and vertically upwards, respectively. \begin{figure}[h] \end{figure} The tension \(\mathbf { T } _ { 1 }\) is \(10 \mathbf { i } + 24 \mathbf { j }\).

1. Calculate the magnitude of \(\mathbf { T } _ { 1 }\) and the angle between \(\mathbf { T } _ { 1 }\) and the vertical. The magnitude of the weight is \(w \mathrm {~N}\).
2. Write down the vector \(\mathbf { W }\) in terms of \(w\) and \(\mathbf { j }\). The tension \(\mathbf { T } _ { 2 }\) is \(k \mathbf { i } + 10 \mathbf { j }\), where \(k\) is a scalar.
3. Find the values of \(k\) and of \(w\).

1. Particle \(P\) has mass \(m\) and particle \(Q\) has mass \(5 m\).

The particles are moving in the same direction along the same straight line on a smooth horizontal surface. Particle \(P\) collides directly with particle \(Q\).
Immediately before the collision, the speed of \(P\) is \(6 u\) and the speed of \(Q\) is \(u\).
Immediately after the collision, the speed of \(P\) is \(x\) and the speed of \(Q\) is \(y\).
The direction of motion of \(P\) is reversed as a result of the collision.
The coefficient of restitution between \(P\) and \(Q\) is \(e\).

1. Find the complete range of possible values of \(e\). Given that \(e = \frac { 3 } { 5 }\)
2. find the total kinetic energy lost in the collision between \(P\) and \(Q\). After the collision, \(Q\) hits a smooth fixed vertical wall that is perpendicular to the direction of motion of \(Q\). Particle \(Q\) rebounds.
   The coefficient of restitution between \(Q\) and the wall is \(f\).
   Given that there is a second collision between \(P\) and \(Q\),
3. find the complete range of possible values of \(f\).

6. [In this question the unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are in a vertical plane, \(\mathbf { i }\) being horizontal and \(\mathbf { j }\) being vertically upwards.] At \(t = 0\) a particle \(P\) is projected from a fixed point \(O\) with velocity ( \(7 \mathbf { i } + 7 \sqrt { 3 } \mathbf { j }\) ) \(\mathrm { m } \mathrm { s } ^ { - 1 }\). The particle moves freely under gravity. The position vector of a point on the path of \(P\) is \(( x \mathbf { i } + y \mathbf { j } ) \mathrm { m }\) relative to \(O\).

1. Show that $$y = \sqrt { 3 } x - \frac { g } { 98 } x ^ { 2 }$$
2. Find the direction of motion of \(P\) when it passes through the point on the path where \(x = 20\) At time \(T\) seconds \(P\) passes through the point with position vector \(( 2 \lambda \mathbf { i } + \lambda \mathbf { j } ) \mathrm { m }\) where \(\lambda\) is a positive constant.
3. Find the value of \(T\).
   DO NOT WIRITE IN THIS AREA

2. A particle of mass 2 kg is moving with velocity \(3 \mathbf { i } \mathrm {~ms} ^ { - 1 }\) when it receives an impulse \(( \lambda \mathbf { i } - 2 \lambda \mathbf { j } )\) Ns, where \(\lambda\) is a constant. Immediately after the impulse is received, the speed of the particle is \(6 \mathrm {~ms} ^ { - 1 }\). Find the possible values of \(\lambda\).

1. \begin{figure}[h] \end{figure} A particle, \(P\), of mass 0.8 kg , moving with speed \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) along a straight line on a smooth horizontal plane, receives a horizontal impulse of magnitude 6 N s. The angle between the initial direction of motion of \(P\) and the direction of the impulse is \(50 ^ { \circ }\), as shown in Figure 1. Find the speed of \(P\) immediately after receiving the impulse.

6. \begin{figure}[h] \end{figure} A cricket ball is hit from a point \(A\) with velocity of \(( p \mathbf { i } + q \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\), at an angle \(\alpha\) above the horizontal. The unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are respectively horizontal and vertically upwards. The point \(A\) is 0.9 m vertically above the point \(O\), which is on horizontal ground. The ball takes 3 seconds to travel from \(A\) to \(B\), where \(B\) is on the ground and \(O B = 57.6 \mathrm {~m}\), as shown in Figure 3. By modelling the motion of the cricket ball as that of a particle moving freely under gravity,

1. find the value of \(p\),
2. show that \(q = 14.4\),
3. find the initial speed of the cricket ball,
4. find the exact value of \(\tan \alpha\).
5. Find the length of time for which the cricket ball is at least 4 m above the ground.
6. State an additional physical factor which may be taken into account in a refinement of the above model to make it more realistic.

1. A particle \(P\) of mass 0.5 kg moves under the action of a single force \(\mathbf { F }\) newtons. At time \(t\) seconds, \(t \geqslant 0 , P\) has velocity \(\mathbf { v } \mathrm { m } \mathrm { s } ^ { - 1 }\), where

$$\mathbf { v } = \left( 4 t - 3 t ^ { 2 } \right) \mathbf { i } + \left( t ^ { 2 } - 8 t - 40 \right) \mathbf { j }$$

1. Find
   1. the magnitude of \(\mathbf { F }\) when \(t = 3\)
   2. the acceleration of \(P\) at the instant when it is moving in the direction of the vector \(- \mathbf { i } - \mathbf { j }\). When \(t = 1 , P\) is at the point \(A\). When \(t = 2 , P\) is at the point \(B\).
2. Find, in terms of \(\mathbf { i }\) and \(\mathbf { j }\), the vector \(\overrightarrow { A B }\).

3. At time \(t\) seconds, a particle \(P\) has position vector \(\mathbf { r }\) metres relative to a fixed origin \(O\), where $$\mathbf { r } = \left( t ^ { 3 } - 3 t \right) \mathbf { i } + 4 t ^ { 2 } \mathbf { j } , t \geq 0$$ Find

1. the velocity of \(P\) at time \(t\) seconds,
2. the time when \(P\) is moving parallel to the vector \(\mathbf { i } + \mathbf { j }\).
   (5)

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 A glass ornament OABCDEFG is a truncated pyramid on a rectangular base (see Fig. 7). All dimensions are in centimetres. \begin{figure}[h] \end{figure}

1. Write down the vectors \(\overrightarrow { \mathrm { CD } }\) and \(\overrightarrow { \mathrm { CB } }\).
2. Find the length of the edge CD.
3. Show that the vector \(4 \mathbf { i } + \mathbf { k }\) is perpendicular to the vectors \(\overrightarrow { \mathrm { CD } }\) and \(\overrightarrow { \mathrm { CB } }\). Hence find the cartesian equation of the plane BCDE .
4. Write down vector equations for the lines OG and AF . Show that they meet at the point P with coordinates (5, 10, 40). You may assume that the lines CD and BE also meet at the point P .
   The volume of a pyramid is \(\frac { 1 } { 3 } \times\) area of base × height.
5. Find the volumes of the pyramids POABC and PDEFG . Hence find the volume of the ornament.

4 The points \(\mathrm { A } , \mathrm { B }\) and C are given by the position vectors \(\mathbf { a } = \binom { - 2 } { 1 } , \mathbf { b } = \binom { 0 } { 5 }\) and \(\mathbf { c } = \binom { 4 } { 3 }\). M is the midpoint of AC .

1. Find the position vector of M .
2. Find the vector \(\overrightarrow { B C }\).
3. Find the position vector of the point D such that \(\overrightarrow { \mathrm { BC } } = \overrightarrow { \mathrm { AD } }\).
4. Show that D lies on BM .

7. Relative to a fixed origin, the points \(A\) and \(B\) have position vectors \(\left( \begin{array} { c } - 4 \\ 1 \\ 3 \end{array} \right)\) and \(\left( \begin{array} { c } - 3 \\ 6 \\ 1 \end{array} \right)\) respectively.

1. Find a vector equation for the line \(l _ { 1 }\) which passes through \(A\) and \(B\). The line \(l _ { 2 }\) has vector equation $$\mathbf { r } = \left( \begin{array} { c } 3 \\ - 7 \\ 9 \end{array} \right) + t \left( \begin{array} { c } 2 \\ - 3 \\ 1 \end{array} \right)$$
2. Show that lines \(l _ { 1 }\) and \(l _ { 2 }\) do not intersect.
3. Find the position vector of the point \(C\) on \(l _ { 2 }\) such that \(\angle A B C = 90 ^ { \circ }\).

7. The line \(l _ { 1 }\) passes through the points \(A\) and \(B\) with position vectors ( \(3 \mathbf { i } + 6 \mathbf { j } - 8 \mathbf { k }\) ) and ( \(8 \mathbf { j } - 6 \mathbf { k }\) ) respectively, relative to a fixed origin.

1. Find a vector equation for \(l _ { 1 }\). The line \(l _ { 2 }\) has vector equation $$\mathbf { r } = ( - 2 \mathbf { i } + 10 \mathbf { j } + 6 \mathbf { k } ) + \mu ( 7 \mathbf { i } - 4 \mathbf { j } + 6 \mathbf { k } ) ,$$ where \(\mu\) is a scalar parameter.
2. Show that lines \(l _ { 1 }\) and \(l _ { 2 }\) intersect.
3. Find the coordinates of the point where \(l _ { 1 }\) and \(l _ { 2 }\) intersect. The point \(C\) lies on \(l _ { 2 }\) and is such that \(A C\) is perpendicular to \(A B\).
4. Find the position vector of \(C\).

6. 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 \(A B D\).

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 vector \(\mathbf { d } - \mathbf { j } + 4 \mathbf { k }\) is perpendicular to the plane through the points \(\mathrm { A } ( 2,0,1 ) , \mathrm { B } ( 1,2,2 )\) and \(\mathrm { C } ( 0 , - 4,1 )\). Hence find the cartesian equation of the plane. [5]

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.

5 The points A , B and C have coordinates \(( 2,0 , - 1 ) , ( 4,3 , - 6 )\) and \(( 9,3 , - 4 )\) respectively.

1. Show that AB is perpendicular to BC .
2. Find the area of triangle ABC .

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.