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

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 \(O A C B\).

3 A particle is in equilibrium under the action of three forces in newtons given by $$\mathbf { F } _ { 1 } = \binom { 8 } { 0 } , \quad \mathbf { F } _ { 2 } = \binom { 2 a } { - 3 a } \quad \text { and } \quad \mathbf { F } _ { 3 } = \binom { 0 } { b } .$$ Find the values of the constants \(a\) and \(b\).

6 Two points, \(A\) and \(B\), have position vectors \(\mathbf { a } = \mathbf { i } - 3 \mathbf { j }\) and \(\mathbf { b } = 4 \mathbf { i } + 3 \mathbf { j }\).
The point C lies on the line \(y = 1\). The lengths of the line segments AC and BC are equal. Determine the position vector of \(C\).

5 You are given that \(\overrightarrow { \mathrm { OA } } = \binom { 3 } { - 1 }\) and \(\overrightarrow { \mathrm { OB } } = \binom { 5 } { - 3 }\). Determine the exact length of \(A B\).

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 \(\frac { 1 } { 2 } \sqrt { 3 } ( 1 + \sqrt { 51 } )\).

10. \begin{figure}[h] \end{figure} Figure 1 shows the parallelogram \(A B C D\).
The points \(A\) and \(B\) have coordinates \(( - 1,3 )\) and \(( 3,4 )\) respectively and lie on the straight line \(l _ { 1 }\).

1. Find an equation for \(l _ { 1 }\), giving your answer in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers. The points \(C\) and \(D\) lie on the straight line \(l _ { 2 }\) which has the equation \(x - 4 y - 21 = 0\).
2. Show that the distance between \(l _ { 1 }\) and \(l _ { 2 }\) is \(k \sqrt { 17 }\), where \(k\) is an integer to be found.
3. Find the area of parallelogram \(A B C D\).

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 \(B\).

5. A straight road passes through villages at the points \(A\) and \(B\) with position vectors ( \(9 \mathbf { i } - 8 \mathbf { j } + 2 \mathbf { k }\) ) and ( \(4 \mathbf { j } + \mathbf { k }\) ) respectively, relative to a fixed origin. The road ends at a junction at the point \(C\) with another straight road which lies along the line with equation $$\mathbf { r } = ( 2 \mathbf { i } + 16 \mathbf { j } - \mathbf { k } ) + \mu ( - 5 \mathbf { i } + 3 \mathbf { j } ) ,$$ where \(\mu\) is a scalar parameter.

1. Find the position vector of \(C\). Given that 1 unit on each coordinate axis represents 200 metres,
2. find the distance, in kilometres, from the village at \(A\) to the junction at \(C\).
   5. 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

6 A motor boat can travel at a speed of \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) relative to the water. It is used to cross a river in which the current flows at \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The resultant velocity of the boat makes an angle of \(60 ^ { \circ }\) to the river bank, as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{eb1f2470-aeeb-4b1d-a6c0-bdeb7048edd5-4_561_1339_1692_350} The angle between the direction in which the boat is travelling relative to the water and the resultant velocity is \(\alpha\).

1. Show that \(\alpha = 16.8 ^ { \circ }\), correct to three significant figures.
2. Find the magnitude of the resultant velocity.

7 The unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are directed east and north respectively. A yacht moves with a constant acceleration. At time \(t\) seconds the position vector of the yacht is \(\mathbf { r }\) metres. When \(t = 0\) the velocity of the yacht is \(( 2 \mathbf { i } - \mathbf { j } ) \mathrm { ms } ^ { - 1 }\), and when \(t = 10\) the velocity of the yacht is \(( - \mathbf { i } + \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\).

1. Find the acceleration of the yacht.
2. When \(t = 0\) the yacht is 20 metres due east of the origin. Find an expression for \(\mathbf { r }\) in terms of \(t\).
3. 1. Show that when \(t = 20\) the yacht is due north of the origin.
   2. Find the speed of the yacht when \(t = 20\).

5 A girl in a boat is rowing across a river, in which the water is flowing at \(0.1 \mathrm {~ms} ^ { - 1 }\). The velocity of the boat relative to the water is \(0.3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and is perpendicular to the bank, as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{965a176a-848c-478d-a748-80fc9dfe4399-4_314_1152_468_450}

1. Find the magnitude of the resultant velocity of the boat.
2. Find the acute angle between the resultant velocity and the bank.
3. The width of the river is 15 metres.
   1. Find the time that it takes the boat to cross the river.
   2. Find the total distance travelled by the boat as it crosses the river.

8 A particle is initially at the origin, where it has velocity \(( 5 \mathbf { i } - 2 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\). It moves with a constant acceleration \(\mathbf { a } \mathrm { ms } ^ { - 2 }\) for 10 seconds to the point with position vector \(75 \mathbf { i }\) metres.

1. Show that \(\mathbf { a } = 0.5 \mathbf { i } + 0.4 \mathbf { j }\).
2. Find the position vector of the particle 8 seconds after it has left the origin.
3. Find the position vector of the particle when it is travelling parallel to the unit vector \(\mathbf { i }\).

2 The velocity of a ship, relative to the water in which it is moving, is \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) due north. The water is moving due east with a speed of \(U \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The resultant velocity of the ship has magnitude \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. Find \(U\).
2. Find the direction of the resultant velocity of the ship. Give your answer as a bearing to the nearest degree.

4 Two particles, \(A\) and \(B\), are moving on a horizontal plane when they collide and coalesce to form a single particle. The mass of \(A\) is 5 kg and the mass of \(B\) is 15 kg . Before the collision, the velocity of \(A\) is \(\left[ \begin{array} { c } 2 U \\ U \end{array} \right] \mathrm { ms } ^ { - 1 }\) and the velocity of \(B\) is \(\left[ \begin{array} { c } V \\ - 1 \end{array} \right] \mathrm { ms } ^ { - 1 }\). After the collision, the velocity of the combined particle is \(\left[ \begin{array} { l } V \\ 0 \end{array} \right] \mathrm { ms } ^ { - 1 }\).

1. Find:
   1. \(U\);
   2. \(V\).
2. Find the speed of \(A\) before the collision.

8 A Jet Ski is at the origin and is travelling due north at \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) when it begins to accelerate uniformly. After accelerating for 40 seconds, it is travelling due east at \(4 \mathrm {~ms} ^ { - 1 }\). The unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are directed east and north respectively.

1. Show that the acceleration of the Jet Ski is \(( 0.1 \mathbf { i } - 0.125 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 2 }\).
2. Find the position vector of the Jet Ski at the end of the 40 second period.
3. The Jet Ski is travelling southeast \(t\) seconds after it leaves the origin.
   1. Find \(t\).
   2. Find the velocity of the Jet Ski at this time.

6 Two forces, \(\mathbf { P } = ( 6 \mathbf { i } - 3 \mathbf { j } )\) newtons and \(\mathbf { Q } = ( 3 \mathbf { i } + 15 \mathbf { j } )\) newtons, act on a particle. The unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are perpendicular.

1. Find the resultant of \(\mathbf { P }\) and \(\mathbf { Q }\).
2. Calculate the magnitude of the resultant of \(\mathbf { P }\) and \(\mathbf { Q }\).
3. When these two forces act on the particle, it has an acceleration of \(( 1.5 \mathbf { i } + 2 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 2 }\). Find the mass of the particle.
4. The particle was initially at rest at the origin.
   1. Find an expression for the position vector of the particle when the forces have been applied to the particle for \(t\) seconds.
   2. Find the distance of the particle from the origin when the forces have been applied to the particle for 2 seconds.

7 A boat is travelling in water that is moving north-east at a speed of \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The velocity of the boat relative to the water is \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) due west. \begin{figure}[h] \end{figure} \begin{figure}[h] \end{figure}

1. Show that the magnitude of the resultant velocity of the boat is \(3.85 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), correct to three significant figures.
2. Find the bearing on which the boat is travelling, giving your answer to the nearest degree.

4 A canoe is paddled across a river which has a width of 20 metres. The canoe moves from the point \(X\) on one bank of the river to the point \(Y\) on the other bank, so that its path is a straight line at an angle \(\alpha\) to the banks. The velocity of the canoe relative to the water is \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) perpendicular to the banks. The water flows at \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) parallel to the banks. \includegraphics[max width=\textwidth, alt={}, center]{f30b02da-a41e-44cb-b45f-9e6a3a9d0528-10_469_1333_543_374} Model the canoe as a particle.

1. Find the magnitude of the resultant velocity of the canoe.
2. Find the angle \(\alpha\).
3. Find the time that it takes for the canoe to travel from \(X\) to \(Y\).
   \includegraphics[max width=\textwidth, alt={}]{f30b02da-a41e-44cb-b45f-9e6a3a9d0528-11_2486_1714_221_153}

5 A particle moves with constant acceleration \(( - 0.4 \mathbf { i } + 0.2 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 2 }\). Initially, it has velocity \(( 4 \mathbf { i } + 0.5 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\). The unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are directed east and north respectively.

1. Find an expression for the velocity of the particle at time \(t\) seconds.
2. 1. Find the velocity of the particle when \(t = 22.5\).
   2. State the direction in which the particle is travelling at this time.
3. Find the time when the speed of the particle is \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

4 A small ferry is used to cross a river which has straight parallel banks that are 200 metres apart. The water in the river moves at a constant speed of \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The ferry travels from a point \(A\) on one bank to a point \(B\) directly opposite \(A\) on the other bank. The velocity of the ferry relative to the water is \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of \(60 ^ { \circ }\) to the upstream bank, as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{d42b2e88-74ea-486b-bb47-f512eb0c185d-4_467_1122_550_459}

1. \(\quad\) Find \(V\).
2. Find the time that it takes for the ferry to cross the river, giving your answer to the nearest second.