Vectors – Year 11 GCSE maths worksheet with answers to download as PDF

\(\mathbf a=\begin{pmatrix}4\\-1\end{pmatrix}\), \(\mathbf b=\begin{pmatrix}-2\\5\end{pmatrix}\) and \(\mathbf c=\begin{pmatrix}3\\3\end{pmatrix}\). Work out each of the following and write your answer as a column vector.

1. \(\mathbf a+\mathbf b\) [1 mark]
2. \(\mathbf a-\mathbf b\) [1 mark]
3. \(3\mathbf c\) [1 mark]
4. \(2\mathbf a+3\mathbf b\) [2 marks]
5. \(\mathbf a-2\mathbf c\) [2 marks]

The magnitude (length) of a vector \(\begin{pmatrix}x\\y\end{pmatrix}\) is \(\sqrt{x^2+y^2}\). Find the magnitude of each vector. Give non-integer answers to 2 decimal places.

1. \(\begin{pmatrix}3\\4\end{pmatrix}\)
2. \(\begin{pmatrix}5\\12\end{pmatrix}\)
3. \(\begin{pmatrix}-8\\6\end{pmatrix}\)
4. \(\begin{pmatrix}2\\5\end{pmatrix}\)
5. \(\begin{pmatrix}-4\\-7\end{pmatrix}\)

Three points have coordinates \(A(1,2)\), \(B(5,5)\) and \(C(-2,4)\).

1. Write down the column vector \(\overrightarrow{AB}\). [1 mark]
2. Work out \(\overrightarrow{BC}\) and \(\overrightarrow{CA}\). [2 marks]
3. Work out \(\overrightarrow{AB}+\overrightarrow{BC}+\overrightarrow{CA}\). What do you notice? [2 marks]
4. Find the length of \(AB\). [1 mark]

Here are four vectors: \(\mathbf p=\begin{pmatrix}2\\3\end{pmatrix}\), \(\mathbf q=\begin{pmatrix}6\\9\end{pmatrix}\), \(\mathbf r=\begin{pmatrix}-4\\-6\end{pmatrix}\) and \(\mathbf s=\begin{pmatrix}4\\5\end{pmatrix}\).

1. Write down two pairs of parallel vectors. [2 marks]
2. Write \(\mathbf q\) and \(\mathbf r\) as multiples of \(\mathbf p\). [2 marks]
3. The vector \(\begin{pmatrix}k\\10\end{pmatrix}\) is parallel to \(\mathbf s\). Find \(k\). [2 marks]

In triangle \(OAB\), \(\overrightarrow{OA}=\mathbf a\) and \(\overrightarrow{OB}=\mathbf b\). \(M\) is the midpoint of \(OA\) and \(N\) is the midpoint of \(OB\).

1. Write \(\overrightarrow{AB}\) in terms of \(\mathbf a\) and \(\mathbf b\). [1 mark]
2. Write \(\overrightarrow{OM}\) and \(\overrightarrow{ON}\). [2 marks]
3. Show that \(\overrightarrow{MN}=\tfrac12\overrightarrow{AB}\). What does this tell you about \(MN\) and \(AB\)? [3 marks]

\(A\) has position vector \(\begin{pmatrix}2\\1\end{pmatrix}\) and \(B\) has position vector \(\begin{pmatrix}8\\10\end{pmatrix}\) relative to the origin \(O\).

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

A drone leaves a warehouse \(W\) and flies to three delivery points. Its movements are given as column vectors in kilometres: the top number is distance east and the bottom number is distance north.

1. Find the vector \(\overrightarrow{WZ}\). [2 marks]
2. How far is \(Z\) from the warehouse? Give your answer to 1 decimal place. [2 marks]
3. The drone flies back from \(Z\) to \(W\) in a straight line. Write down the vector for the return flight. [1 mark]
4. The drone travels at 30 km/h. How many minutes does the return flight take? [2 marks]

A group of hikers walks from a campsite \(C\) to a lake \(L\) and then to a hut \(H\). Using east and north as the directions, \(\overrightarrow{CL}=\begin{pmatrix}4\\3\end{pmatrix}\) km and \(\overrightarrow{LH}=\begin{pmatrix}2\\-6\end{pmatrix}\) km.

1. Work out \(\overrightarrow{CH}\). [2 marks]
2. Work out the distance \(CL\), the distance \(LH\) and the total distance walked. Give answers to 2 decimal places. [3 marks]
3. How much shorter is the direct route from \(C\) to \(H\) than the route through the lake? [3 marks]

A treasure map says: "Start at the old oak \(O\). The chest is at \(T\) where \(\overrightarrow{OT}=\begin{pmatrix}12\\5\end{pmatrix}\) paces. The key is at \(K\) where \(\overrightarrow{OK}=\begin{pmatrix}-4\\8\end{pmatrix}\) paces."

1. Find \(\overrightarrow{KT}\). [2 marks]
2. How many paces is it from the key to the chest? Give your answer to the nearest pace. [2 marks]
3. A second chest is at the midpoint of \(KT\). Find its position vector from \(O\). [2 marks]
4. Mira walks directly from \(O\) to \(T\). Show that she walks 13 paces. [1 mark]

A ferry sails across a river. Without a current it would travel \(\begin{pmatrix}0\\6\end{pmatrix}\) km (east, north) in one hour. The current pushes it by \(\begin{pmatrix}3\\0\end{pmatrix}\) km in that hour.

1. Work out the actual displacement of the ferry after one hour. [2 marks]
2. Find the ferry\'s actual speed in km/h to 1 decimal place. [2 marks]
3. The ferry starts at a landing stage \(S\). How far east of the landing stage on the opposite bank does it arrive after crossing 6 km north? [2 marks]
4. The captain wants to arrive directly opposite \(S\). What vector must she aim for so that the current cancels the sideways motion? [2 marks]

A robot moves on a square grid. One step is \(\mathbf x=\begin{pmatrix}2\\1\end{pmatrix}\) and another step is \(\mathbf y=\begin{pmatrix}-1\\3\end{pmatrix}\). The robot starts at the origin.

1. Where is the robot after doing step \(\mathbf x\) three times and step \(\mathbf y\) twice? [2 marks]
2. Find the position after \(4\mathbf x-\mathbf y\). [2 marks]
3. The robot must reach the point \((7,7)\). It does step \(\mathbf x\) \(m\) times and step \(\mathbf y\) \(n\) times. Write two equations and solve them to find \(m\) and \(n\). [4 marks]

A small plane flies on a heading that would take it \(\begin{pmatrix}120\\50\end{pmatrix}\) km in one hour. A wind blows it by \(\begin{pmatrix}-30\\10\end{pmatrix}\) km per hour. The plane flies for 3 hours.

1. Find the resultant velocity of the plane over the ground. [2 marks]
2. Find the position of the plane after 3 hours relative to its starting point. [2 marks]
3. How far from the start is it? Give your answer to the nearest km. [2 marks]
4. The destination is at \(\begin{pmatrix}270\\180\end{pmatrix}\) km. Is the plane on course after 3 hours? Explain. [2 marks]

On a pitch plan, a midfielder \(M\) is at \((10,20)\) metres. A striker \(S\) is at \((34,52)\) and a winger \(W\) is at \((50,20)\).

1. Write down the vectors \(\overrightarrow{MS}\) and \(\overrightarrow{SW}\). [2 marks]
2. How long is each pass, to 1 decimal place? [3 marks]
3. A ball travels at 16 m/s. How long does the pass \(M\) to \(S\) take? [2 marks]
4. Is it shorter to pass directly from \(M\) to \(W\), or via \(S\)? Show your working. [2 marks]

In an orienteering event, runners visit three checkpoints. Relative to the start \(S\), checkpoint \(A\) is at \(\begin{pmatrix}300\\400\end{pmatrix}\) m, checkpoint \(B\) is at \(\begin{pmatrix}900\\400\end{pmatrix}\) m and checkpoint \(C\) is at \(\begin{pmatrix}900\\1000\end{pmatrix}\) m (east, north).

1. Write down the vectors \(\overrightarrow{AB}\) and \(\overrightarrow{BC}\). [2 marks]
2. Show that triangle \(ABC\) has a right angle at \(B\). [2 marks]
3. Work out the length of the course \(S\to A\to B\to C\). [3 marks]
4. A runner cuts straight from \(A\) to \(C\). How much shorter is this than going via \(B\)? Give your answer to the nearest metre. [3 marks]

\(OABC\) is a parallelogram. \(\overrightarrow{OA}=\mathbf a\) and \(\overrightarrow{OC}=\mathbf c\). \(M\) is the midpoint of \(AB\) and \(N\) is the midpoint of \(BC\).

1. Write \(\overrightarrow{OB}\) in terms of \(\mathbf a\) and \(\mathbf c\). [1 mark]
2. Write \(\overrightarrow{OM}\) and \(\overrightarrow{ON}\). [3 marks]
3. Write \(\overrightarrow{MN}\) and show that \(MN\) is parallel to \(AC\). [3 marks]
4. What fraction of the length of \(AC\) is \(MN\)? [1 mark]

\(\overrightarrow{OA}=\mathbf a\) and \(\overrightarrow{OB}=\mathbf b\). Point \(P\) divides \(AB\) so that \(AP:PB=2:3\). Point \(Q\) is on \(OB\) with \(OQ:QB=2:3\).

1. Find \(\overrightarrow{AB}\). [1 mark]
2. Show that \(\overrightarrow{OP}=\tfrac35\mathbf a+\tfrac25\mathbf b\). [3 marks]
3. Find \(\overrightarrow{OQ}\) and \(\overrightarrow{PQ}\). [3 marks]
4. What do you notice about \(\overrightarrow{PQ}\) and \(\overrightarrow{OA}\)? Explain what this means for the lines. [2 marks]

\(\overrightarrow{OA}=\mathbf a\) and \(\overrightarrow{OB}=\mathbf b\). \(\overrightarrow{OP}=2\mathbf a+\mathbf b\), \(\overrightarrow{OQ}=5\mathbf a+4\mathbf b\) and \(\overrightarrow{OR}=-\mathbf a-2\mathbf b\).

1. Find \(\overrightarrow{PQ}\) and \(\overrightarrow{PR}\). [3 marks]
2. Show that \(P\), \(Q\) and \(R\) lie on a straight line. [3 marks]
3. What is special about the position of \(P\) on the line \(QR\)? [2 marks]

A coastguard station \(C\) is at the origin. A boat \(B\) is at \(\begin{pmatrix}12\\16\end{pmatrix}\) km and a lifeboat \(L\) is at \(\begin{pmatrix}-6\\8\end{pmatrix}\) km. A helicopter \(H\) flies from \(C\) to the midpoint of \(BL\).

1. Find \(\overrightarrow{BL}\). [2 marks]
2. Find the position vector of the midpoint \(M\) of \(BL\). [2 marks]
3. How far does the helicopter fly from \(C\) to \(M\)? Give your answer to 1 decimal place. [2 marks]
4. The helicopter flies at 150 km/h. Work out the time taken in minutes. [2 marks]
5. A second vessel is at \(V\) with \(\overrightarrow{CV}=2\overrightarrow{CM}\). Find the coordinates of \(V\) and explain where \(M\) is on \(CV\). [2 marks]

\(OABC\) is a trapezium with \(OA\) parallel to \(CB\) and \(CB=2OA\). \(\overrightarrow{OA}=\mathbf a\) and \(\overrightarrow{OC}=\mathbf c\). \(D\) is the midpoint of \(AB\).

1. Write \(\overrightarrow{CB}\) and \(\overrightarrow{AB}\) in terms of \(\mathbf a\) and \(\mathbf c\). [3 marks]
2. Find \(\overrightarrow{OD}\). [2 marks]
3. \(E\) is on \(OB\) with \(OE:EB=1:2\). Find \(\overrightarrow{OE}\) and show that \(D\), \(E\) and \(C\) are not in a straight line unless a certain relation holds. [3 marks]

In triangle \(OAB\), \(\overrightarrow{OA}=\mathbf a\) and \(\overrightarrow{OB}=\mathbf b\). \(M\) is the midpoint of \(AB\). \(G\) is the point on \(OM\) such that \(OG:GM=2:1\).

1. Find \(\overrightarrow{OM}\) and \(\overrightarrow{OG}\). [3 marks]
2. \(N\) is the midpoint of \(OA\). Show that \(B\), \(G\) and \(N\) lie on a straight line and find the ratio \(BG:GN\). [5 marks]
3. Explain why this shows that the three medians of a triangle meet at one point. [2 marks]