Vectors in two dimensions – Year 12 A level maths worksheet with answers to download as PDF

\(\mathbf a=3\mathbf i-2\mathbf j\) and \(\mathbf b=-\mathbf i+5\mathbf j\).

1. Find \(\mathbf a+\mathbf b\). [1 mark]
2. Find \(2\mathbf a-3\mathbf b\). [2 marks]
3. Find \(|\mathbf a|\) as a surd. [2 marks]
4. Find a unit vector in the direction of \(\mathbf b\). [2 marks]

Give angles to 1 decimal place.

1. Find the magnitude of \(-5\mathbf i+12\mathbf j\) and the angle it makes with the positive \(x\)-axis (measured anticlockwise). [4 marks]
2. Find the magnitude of \(4\mathbf i-3\mathbf j\) and its direction as an angle between \(0^\circ\) and \(360^\circ\). [4 marks]
3. Write, in the form \(p\mathbf i+q\mathbf j\), the vector of magnitude 10 at \(30^\circ\) above the positive \(x\)-axis. [2 marks]

Work with parallel vectors.

1. \(\mathbf p=3\mathbf i+\lambda\mathbf j\) is parallel to \(\mathbf q=-6\mathbf i+8\mathbf j\). Find \(\lambda\). [3 marks]
2. Show that \(\begin{pmatrix}4\\-6\end{pmatrix}\) and \(\begin{pmatrix}-6\\9\end{pmatrix}\) are parallel and state the ratio of their magnitudes. [3 marks]
3. \(\mathbf a=2\mathbf i+k\mathbf j\) and \(\mathbf b=(k+1)\mathbf i+6\mathbf j\) are parallel. Find the possible values of \(k\). [4 marks]

Relative to the origin \(O\), \(A\) has position vector \(2\mathbf i+5\mathbf j\) and \(B\) has position vector \(8\mathbf i-\mathbf j\).

1. Find \(\overrightarrow{AB}\) and \(|\overrightarrow{AB}|\). [3 marks]
2. Find the position vector of the midpoint \(M\). [2 marks]
3. \(P\) lies on \(AB\) with \(AP:PB=1:2\). Find the position vector of \(P\). [3 marks]

\(OABC\) is a parallelogram with \(\overrightarrow{OA}=\mathbf a\) and \(\overrightarrow{OC}=\mathbf c\).

1. Write \(\overrightarrow{OB}\) and \(\overrightarrow{AC}\) in terms of \(\mathbf a\) and \(\mathbf c\). [2 marks]
2. Find the position vectors of the midpoints of \(OB\) and \(AC\). [3 marks]
3. What does your answer show about the diagonals of a parallelogram? [1 mark]

A boat has velocity \(3\mathbf i+4\mathbf j\) m/s, where \(\mathbf i\) is east and \(\mathbf j\) is north.

1. Find its speed and its bearing to 1 decimal place. [3 marks]
2. Find its position after 10 s if it starts at the origin. [1 mark]
3. A second vessel travels at 12 km/h on a bearing of \(120^\circ\). Write its velocity in the form \(a\mathbf i+b\mathbf j\) to 2 decimal places. [3 marks]

An orienteering course in a forest starts at \(S(0,0)\), goes to control \(C_1(3,4)\), then \(C_2(8,1)\) and finishes at \(F(8,-5)\). Coordinates are in kilometres east and north of the start.

1. Write down the vector for each leg. [3 marks]
2. Find the total distance run to 2 decimal places. [3 marks]
3. Find the straight-line distance from \(S\) to \(F\). [2 marks]
4. Find the bearing of \(F\) from \(S\) to the nearest degree. [2 marks]

A river is 120 m wide and flows east at 3 m/s. A boat can travel at 4 m/s in still water. Let \(\mathbf i\) be east and \(\mathbf j\) be north (across the river).

1. The boat steers due north. Find its velocity over the ground, the time to cross, and how far downstream it lands. [4 marks]
2. Find the bearing of the boat\'s actual path. [2 marks]
3. To land directly opposite its starting point the boat steers at an angle \(\alpha\) west of north. Find \(\alpha\). [3 marks]
4. Find the time to cross in this case. [3 marks]

An aeroplane has an airspeed of 240 km/h. There is a wind of 50 km/h blowing towards the east. Take \(\mathbf i\) as east and \(\mathbf j\) as north.

1. The pilot heads due north. Find the ground velocity, the ground speed and the bearing of the path. [4 marks]
2. To fly due north over the ground the pilot heads at a bearing \(\beta\). Find \(\beta\) to 1 decimal place. [3 marks]
3. Find the ground speed in this case. [2 marks]
4. How long does a 600 km flight take in this case? Give your answer in hours and minutes. [2 marks]

Two drones fly in a field. After \(t\) seconds, drone \(P\) is at \(\mathbf r_P=(2+3t)\mathbf i+(1+2t)\mathbf j\) and drone \(Q\) is at \(\mathbf r_Q=(14-t)\mathbf i+(-3+4t)\mathbf j\) (metres).

1. Find the velocity of each drone. [2 marks]
2. Find their positions when \(t=2\) and the distance between them. [3 marks]
3. Show that the drones never collide. [4 marks]
4. At what time are the two drones at the same \(x\)-coordinate? [1 mark]

A zip wire runs from the top of a tower \(A\) at \((-4,6)\) to a platform tower \(B\) at \((6,-4)\) on a plan (units: tens of metres). A rest platform \(P\) divides \(AB\) in the ratio \(AP:PB=3:2\).

1. Find \(\overrightarrow{AB}\). [1 mark]
2. Find the position vector of \(P\). [3 marks]
3. Find the length of the whole wire in metres to the nearest metre. [2 marks]
4. The wire costs £48 per metre. Find the cost. [2 marks]

On a football pitch plan (metres), a player passes the ball from \(A(10,8)\) to \(B(22,17)\) and then to \(C(17,29)\). The ball travels at 15 m/s.

1. Write down \(\overrightarrow{AB}\) and \(\overrightarrow{BC}\). [2 marks]
2. Find the length of each pass. [3 marks]
3. Find the total time for the two passes. [2 marks]
4. How much longer is the route \(A\to B\to C\) than the direct route \(A\to C\)? [3 marks]

Three pylons of a cable car are at \(A(1,2)\), \(B(4,8)\) and \(C(10,20)\) on a map (units: 100 m).

1. Find \(\overrightarrow{AB}\) and \(\overrightarrow{BC}\). [2 marks]
2. Show that \(A\), \(B\) and \(C\) are collinear. [2 marks]
3. Find the ratio \(AB:BC\). [1 mark]
4. The cable \(AC\) costs £18 per metre. Find the length and cost of the cable. [3 marks]

Two ships leave port \(O\) at the same time. Ship \(A\) travels at 24 km/h on a bearing of \(060^\circ\). Ship \(B\) travels at 18 km/h on a bearing of \(150^\circ\).

1. Write the position of each ship after 2 hours as a vector in \(\mathbf i,\mathbf j\) form (east, north) to 1 decimal place. [4 marks]
2. Find the distance between the ships after 2 hours. [3 marks]
3. Find the bearing of \(B\) from \(A\) at this time. [3 marks]

\(OABC\) is a parallelogram with \(\overrightarrow{OA}=\mathbf a\) and \(\overrightarrow{OC}=\mathbf c\). \(M\) is the point on \(AB\) with \(AM:MB=1:2\) and \(N\) is the point on \(CB\) with \(CN:NB=1:2\).

1. Find \(\overrightarrow{OM}\) and \(\overrightarrow{ON}\) in terms of \(\mathbf a\) and \(\mathbf c\). [4 marks]
2. Show that \(\overrightarrow{MN}=\tfrac23\overrightarrow{AC}\). [3 marks]
3. What does this tell you about the lines \(MN\) and \(AC\)? [2 marks]
4. Triangles \(BMN\) and \(BAC\) are similar. State the scale factor and how the answer to part (b) confirms it. [2 marks]

\(\overrightarrow{OP}=2\mathbf a+\mathbf b\), \(\overrightarrow{OQ}=5\mathbf a+4\mathbf b\) and \(\overrightarrow{OR}=8\mathbf a+7\mathbf b\), where \(\mathbf a\) and \(\mathbf b\) are not parallel.

1. Find \(\overrightarrow{PQ}\) and \(\overrightarrow{PR}\). [3 marks]
2. Show that \(P\), \(Q\) and \(R\) lie on a straight line. [3 marks]
3. Find the ratio \(PQ:QR\). [2 marks]
4. The point \(S\) has \(\overrightarrow{OS}=11\mathbf a+k\mathbf b\) and lies on the same line. Find \(k\). [3 marks]

A boat has position vector \(\mathbf r=(2+3t)\mathbf i+(-1+2t)\mathbf j\) km at time \(t\) hours. A lighthouse \(L\) stands at \((10,0)\).

1. Find the speed of the boat. [2 marks]
2. Show that \(\overrightarrow{LB}\) has squared length \(13t^2-52t+65\) where \(B\) is the boat\'s position. [4 marks]
3. Find the time of closest approach to the lighthouse and the shortest distance. [4 marks]
4. At what time does the boat cross the \(x\)-axis (the coastline)? [2 marks]

A hiker walks three legs. Leg 1 is 5 km on a bearing of \(030^\circ\), leg 2 is 8 km on a bearing of \(120^\circ\) and leg 3 is 6 km on a bearing of \(250^\circ\).

1. Check the entries for leg 3 and add the missing totals. [3 marks]
2. Find the total displacement vector. [2 marks]
3. How far is the hiker from the start? Give your answer to 2 decimal places. [2 marks]
4. On what bearing must the hiker walk to return directly? [3 marks]

\(ABCD\) is any quadrilateral with position vectors \(\mathbf a\), \(\mathbf b\), \(\mathbf c\) and \(\mathbf d\). \(P\), \(Q\), \(R\) and \(S\) are the midpoints of \(AB\), \(BC\), \(CD\) and \(DA\).

1. Write the position vectors of \(P\), \(Q\), \(R\) and \(S\). [2 marks]
2. Show that \(\overrightarrow{PQ}=\tfrac12(\mathbf c-\mathbf a)\) and \(\overrightarrow{SR}=\tfrac12(\mathbf c-\mathbf a)\). [4 marks]
3. Prove that \(PQRS\) is a parallelogram. [2 marks]
4. If \(ABCD\) is a rectangle, what special parallelogram is \(PQRS\)? Justify. [3 marks]

Aircraft \(P\) is at the origin at time \(t=0\) and flies with velocity \(200\mathbf i+150\mathbf j\) km/h. Aircraft \(Q\) is at \(300\mathbf i\) km at \(t=0\) and flies with velocity \(-100\mathbf i+140\mathbf j\) km/h.

1. Write \(\mathbf r_P\) and \(\mathbf r_Q\) at time \(t\) hours. [2 marks]
2. Show that \(\overrightarrow{PQ}=(300-300t)\mathbf i-10t\mathbf j\). [2 marks]
3. The separation rule requires at least 15 km. Show that the squared distance is \(90100t^2-180000t+90000\) and find its minimum to the nearest km. [5 marks]
4. State whether the rule is broken. [1 mark]

A lifeboat leaves harbour \(H\) at the origin at 12:00 and travels with constant velocity \(\mathbf v\) km/h to intercept a yacht. At 12:00 the yacht is at \((12,16)\) km and moves with constant velocity \((-3\mathbf i-2\mathbf j)\) km/h. The lifeboat\'s speed is 10 km/h.

1. Write the position of the yacht at time \(t\). [1 mark]
2. The lifeboat reaches the yacht at time \(T\). Show that \(T\) satisfies an equation of the form \(aT^2+bT+c=0\) and find \(a\), \(b\) and \(c\). [5 marks]
3. Solve it to find the interception time to the nearest minute and the position of the yacht. [4 marks]