How do I find the vector between two points?

Subtract the start coordinates from the end coordinates. The vector from A(1,2) to B(5,5) is (4,3).

How do I find the length of a vector?

Use Pythagoras: square the two numbers, add and square root. The length of (3,4) is 5.

How do I prove three points are in a straight line?

Show two of the vectors between them are multiples of each other and share a point. This is Higher tier.

What is a position vector?

It is the vector from the origin O to a point, for example OA = a.

Year 11 vectors revision notes to download as PDF

This Year 11 vectors revision notes is written for pupils aged 15-16 and follows the National Curriculum. In short, it shows you how to use column vectors and vector diagrams. Before you start, it helps to revise similarity and congruence and circle theorems, because this lesson builds on them.

Key points

A vector has magnitude and direction; a column vector \(\begin{pmatrix}x\\y\end{pmatrix}\) means \(x\) across and \(y\) up, and \(\overrightarrow{AB}\) is the movement from \(A\) to \(B\).
Add and subtract vectors by adding and subtracting the top numbers and the bottom numbers, and multiply by a scalar \(k\) by multiplying both numbers: \(2\mathbf a-3\mathbf b\).
The magnitude of \(\begin{pmatrix}x\\y\end{pmatrix}\) is \(\sqrt{x^2+y^2}\) from Pythagoras; for example \(\begin{pmatrix}5\\12\end{pmatrix}\) has length 13.
\(\overrightarrow{AB}=\mathbf b-\mathbf a\) (end minus start) and \(\overrightarrow{BA}=-\overrightarrow{AB}\); a closed journey adds to the zero vector.
Parallel vectors are scalar multiples of each other, for example \(\begin{pmatrix}2\\3\end{pmatrix}\) and \(\begin{pmatrix}6\\9\end{pmatrix}\) are parallel because \(\mathbf q=3\mathbf p\).
The midpoint of \(AB\) has position vector \(\tfrac12(\mathbf a+\mathbf b)\); if \(AP:PB=m:n\) then \(\overrightarrow{AP}=\dfrac{m}{m+n}\overrightarrow{AB}\).

Table of Contents

Year 11 vectors revision notes: what you will learn

Column vectors and vector arithmetic are on both tiers. Geometric arguments and proofs with vectors are Higher tier (marked Higher).

Starter activity

A treasure map says: "Walk 4 paces east and 3 paces north, then 2 paces west and 5 paces north." Where are you now? Draw the route on squared paper. Can you find a single walk from the start that gets you to the same place? How long is that walk? Think about how you could write each walk as a pair of numbers.

Writing vectors

Definition: vector and scalar

A vector has both a size (magnitude) and a direction. A scalar has only size. A column vector \(\begin{pmatrix}x\\y\end{pmatrix}\) means "\(x\) across, \(y\) up". In diagrams a vector is written \(\overrightarrow{AB}\) (from \(A\) to \(B\)) or as a bold letter \(\mathbf a\); in handwriting, write a letter with a wavy line underneath, \(\underset{\sim}{a}\).

Key fact: the negative of a vector

The vector \(\overrightarrow{BA}\) is the negative of \(\overrightarrow{AB}\): \(\overrightarrow{BA}=-\overrightarrow{AB}\), the same length in the opposite direction.

Adding, subtracting and multiplying

Rule: vector arithmetic

Add or subtract the top numbers and the bottom numbers separately.
Multiply a vector by a scalar by multiplying both numbers.

Worked example 1: arithmetic

\(\mathbf a=\begin{pmatrix}3\\2\end{pmatrix}\), \(\mathbf b=\begin{pmatrix}1\\-4\end{pmatrix}\). Find \(\mathbf a+\mathbf b\), \(\mathbf a-\mathbf b\) and \(2\mathbf a-3\mathbf b\).

\(\mathbf a+\mathbf b=\begin{pmatrix}4\\-2\end{pmatrix}\), \(\mathbf a-\mathbf b=\begin{pmatrix}2\\6\end{pmatrix}\) and \(2\mathbf a-3\mathbf b=\begin{pmatrix}6\\4\end{pmatrix}-\begin{pmatrix}3\\-12\end{pmatrix}=\begin{pmatrix}3\\16\end{pmatrix}\).

Magnitude and vectors between points

Rule: magnitude and \(\overrightarrow{AB}\)

The length of \(\begin{pmatrix}x\\y\end{pmatrix}\) is found with Pythagoras: \(\sqrt{x^2+y^2}\). The vector from \(A(x_1,y_1)\) to \(B(x_2,y_2)\) is \(\overrightarrow{AB}=\begin{pmatrix}x_2-x_1\\y_2-y_1\end{pmatrix}\) (end minus start).

Worked example 2: magnitude

Find the magnitude of \(\begin{pmatrix}5\\12\end{pmatrix}\), and the vector and length from \(A(1,2)\) to \(B(5,5)\).

The magnitude of \(\begin{pmatrix}5\\12\end{pmatrix}\) is \(\sqrt{25+144}=13\). For \(A(1,2)\) and \(B(5,5)\), \(\overrightarrow{AB}=\begin{pmatrix}4\\3\end{pmatrix}\) and its length is 5.

Vector journeys and parallel vectors

Method: vector journeys

Travel from one point to another along any route made of known vectors, adding the vectors along the way.
To go backwards along a vector, use its negative.
Simplify by collecting like terms.

Worked example 3: triangle \(OAB\)

\(\overrightarrow{OA}=\mathbf a\) and \(\overrightarrow{OB}=\mathbf b\). Find \(\overrightarrow{AB}\), the midpoint \(M\) of \(AB\), and \(P\) on \(AB\) with \(AP:PB=1:2\).

\(\overrightarrow{AB}=\overrightarrow{AO}+\overrightarrow{OB}=-\mathbf a+\mathbf b=\mathbf b-\mathbf a\). \(\overrightarrow{OM}=\mathbf a+\tfrac12(\mathbf b-\mathbf a)=\tfrac12(\mathbf a+\mathbf b)\). For \(P\), \(\overrightarrow{AP}=\tfrac13\overrightarrow{AB}\), so \(\overrightarrow{OP}=\mathbf a+\tfrac13(\mathbf b-\mathbf a)=\tfrac23\mathbf a+\tfrac13\mathbf b\).

Definition: parallel vectors

Two vectors are parallel if one is a scalar multiple of the other. \(\begin{pmatrix}6\\-9\end{pmatrix}=3\begin{pmatrix}2\\-3\end{pmatrix}\), so they are parallel. \(\begin{pmatrix}2\\5\end{pmatrix}\) and \(\begin{pmatrix}6\\12\end{pmatrix}\) are not, because \(6\div2=3\) but \(12\div5\ne3\). In algebra, \(2\mathbf a+4\mathbf b\) is parallel to \(\mathbf a+2\mathbf b\).

Worked example 4: a parallelogram

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

\(\overrightarrow{OB}=\mathbf a+\mathbf c\) and \(\overrightarrow{AC}=\mathbf c-\mathbf a\).

Position vectors and ratios

Definition: position vector

A position vector tells you where a point is, measured from the origin \(O\). A ratio \(1:2\) on a line means \(\tfrac13\) of the whole line, because \(1+2=3\) parts.

Worked example 5: dividing a line in a ratio

\(P\) and \(Q\) have position vectors \(\begin{pmatrix}2\\3\end{pmatrix}\) and \(\begin{pmatrix}8\\0\end{pmatrix}\). Find \(R\) on \(PQ\) with \(PR:RQ=1:2\).

\(\overrightarrow{PQ}=\begin{pmatrix}6\\-3\end{pmatrix}\). \(\overrightarrow{PR}=\tfrac13\overrightarrow{PQ}=\begin{pmatrix}2\\-1\end{pmatrix}\) and \(\overrightarrow{OR}=\begin{pmatrix}2\\3\end{pmatrix}+\begin{pmatrix}2\\-1\end{pmatrix}=\begin{pmatrix}4\\2\end{pmatrix}\).

Worked example 6: a resultant and simplifying

A plane flies with velocity \(\begin{pmatrix}200\\0\end{pmatrix}\) km/h in still air and the wind has velocity \(\begin{pmatrix}-30\\40\end{pmatrix}\) km/h. Find the resultant speed. Simplify \(3(\mathbf a+2\mathbf b)-2(\mathbf a-\mathbf b)\).

The resultant is \(\begin{pmatrix}170\\40\end{pmatrix}\), with speed \(\sqrt{170^2+40^2}=\sqrt{30\,500}=174.6\) km/h. Collect like terms: \(3\mathbf a+6\mathbf b-2\mathbf a+2\mathbf b=\mathbf a+8\mathbf b\).

(Higher) Proofs with vectors

Method (Higher): writing a vector proof

Set out the vectors you know with letters.
Find the vectors you need by travelling along known routes.
Simplify and compare.
Write a conclusion in words: "so \(MN\) is parallel to \(AB\)" or "so \(A\), \(B\) and \(C\) lie on a straight line".

To prove a straight line, you need two statements: the vectors are parallel, and they share a common point.

Worked example 7 (Higher): collinear points

Show that \(A(1,1)\), \(B(3,2)\), \(C(7,4)\) are collinear.

\(\overrightarrow{AB}=\begin{pmatrix}2\\1\end{pmatrix}\), \(\overrightarrow{AC}=\begin{pmatrix}6\\3\end{pmatrix}=3\overrightarrow{AB}\). The vectors are parallel and share the point \(A\), so \(A\), \(B\), \(C\) are collinear.

Worked example 8 (Higher): diagonals of a parallelogram

\(ABCD\) is a parallelogram with \(\overrightarrow{AB}=\mathbf p\) and \(\overrightarrow{AD}=\mathbf q\). Prove that the diagonals bisect each other.

\(\overrightarrow{AC}=\mathbf p+\mathbf q\) and \(\overrightarrow{BD}=\mathbf q-\mathbf p\). Let \(M\) be the midpoint of \(AC\): \(\overrightarrow{AM}=\tfrac12(\mathbf p+\mathbf q)\). Then \(\overrightarrow{BM}=\overrightarrow{BA}+\overrightarrow{AM}=-\mathbf p+\tfrac12\mathbf p+\tfrac12\mathbf q=\tfrac12(\mathbf q-\mathbf p)=\tfrac12\overrightarrow{BD}\). So \(M\) is also the midpoint of \(BD\).

Worked example 9 (Higher): the midpoint theorem

In triangle \(OAB\), \(M\) and \(N\) are the midpoints of \(OA\) and \(OB\). Show that \(MN\) is parallel to \(AB\).

\(\overrightarrow{MN}=\overrightarrow{MO}+\overrightarrow{ON}=-\tfrac12\mathbf a+\tfrac12\mathbf b=\tfrac12(\mathbf b-\mathbf a)=\tfrac12\overrightarrow{AB}\). So \(MN\) is parallel to \(AB\) and half its length.

Watch out: common mistakes

Getting the sign wrong in \(\overrightarrow{AB}=\mathbf b-\mathbf a\) (end minus start).
Adding magnitudes instead of the vectors.
Writing a vector as a coordinate pair \((3,2)\) when a column vector is needed.
Stopping a proof without a conclusion sentence.

Worked example 10: Resultant of two journeys

Anna walks \(\begin{pmatrix}5\\2\end{pmatrix}\) km (east, north) and then \(\begin{pmatrix}-1\\6\end{pmatrix}\) km. How far is she from the start, and what is the bearing of the start from her?

Add the vectors: \(\begin{pmatrix}5\\2\end{pmatrix}+\begin{pmatrix}-1\\6\end{pmatrix}=\begin{pmatrix}4\\8\end{pmatrix}\). The distance is \(\sqrt{4^2+8^2}=\sqrt{80}=8.94\) km. For a bearing, draw a sketch: she is 4 km east and 8 km north of the start. The start is therefore south-west of her. The angle between south and the line is \(\tan^{-1}\left(\tfrac48\right)=26.6^\circ\), so the bearing of the start from Anna is \(180^\circ+26.6^\circ=206.6^\circ\).

Worked example 11: Showing that a quadrilateral is a parallelogram

\(A(1,1)\), \(B(6,2)\), \(C(8,6)\) and \(D(3,5)\). Show that \(ABCD\) is a parallelogram.

Find \(\overrightarrow{AB}=\begin{pmatrix}5\\1\end{pmatrix}\) and \(\overrightarrow{DC}=\begin{pmatrix}5\\1\end{pmatrix}\). They are equal, so \(AB\) and \(DC\) are parallel and equal in length. A quadrilateral with one pair of opposite sides equal and parallel is a parallelogram. To be thorough, also check \(\overrightarrow{AD}=\begin{pmatrix}2\\4\end{pmatrix}=\overrightarrow{BC}=\begin{pmatrix}2\\4\end{pmatrix}\).

Worked example 12: Scalar multiples and parallel lines

\(\overrightarrow{OA}=\begin{pmatrix}1\\2\end{pmatrix}\) and \(\overrightarrow{OB}=\begin{pmatrix}7\\5\end{pmatrix}\). \(\overrightarrow{OC}=\begin{pmatrix}4\\2\end{pmatrix}\). Show that \(AB\) is parallel to \(OC\).

\(\overrightarrow{AB}=\begin{pmatrix}7\\5\end{pmatrix}-\begin{pmatrix}1\\2\end{pmatrix}=\begin{pmatrix}6\\3\end{pmatrix}\). This equals \(\tfrac32\times\begin{pmatrix}4\\2\end{pmatrix}\), since \(\tfrac32\times4=6\) and \(\tfrac32\times2=3\). A multiple of \(\overrightarrow{OC}\) means the lines are parallel. Also \(AB:OC=3:2\).

Worked example 13: Midpoint using position vectors

\(A\) has position vector \(\begin{pmatrix}-2\\4\end{pmatrix}\) and \(B\) has position vector \(\begin{pmatrix}6\\0\end{pmatrix}\). Find the midpoint \(M\) and the point \(P\) on \(AB\) with \(AP:PB=3:1\).

\(\overrightarrow{OM}=\tfrac12(\mathbf a+\mathbf b)=\tfrac12\begin{pmatrix}4\\4\end{pmatrix}=\begin{pmatrix}2\\2\end{pmatrix}\). For \(P\): \(\overrightarrow{AB}=\begin{pmatrix}8\\-4\end{pmatrix}\) and \(AP=\tfrac34AB\), so \(\overrightarrow{OP}=\begin{pmatrix}-2\\4\end{pmatrix}+\tfrac34\begin{pmatrix}8\\-4\end{pmatrix}=\begin{pmatrix}-2\\4\end{pmatrix}+\begin{pmatrix}6\\-3\end{pmatrix}=\begin{pmatrix}4\\1\end{pmatrix}\).

Worked example 14: Magnitude with a scalar multiple

\(\mathbf a=\begin{pmatrix}3\\-4\end{pmatrix}\). Work out \(|3\mathbf a|\) and compare it with \(|\mathbf a|\).

\(3\mathbf a=\begin{pmatrix}9\\-12\end{pmatrix}\), so \(|3\mathbf a|=\sqrt{81+144}=\sqrt{225}=15\). \(|\mathbf a|=\sqrt{9+16}=5\). So \(|3\mathbf a|=3|\mathbf a|\): multiplying a vector by a positive scalar \(k\) multiplies its length by \(k\) and keeps its direction. A negative scalar reverses the direction as well.

Worked example 15: A full vector proof (Higher)

\(OABC\) is a parallelogram with \(\overrightarrow{OA}=\mathbf a\) and \(\overrightarrow{OC}=\mathbf c\). Prove that the diagonals \(OB\) and \(AC\) bisect each other.

The diagonal \(OB\) has vector \(\mathbf a+\mathbf c\) and the diagonal \(AC\) has vector \(\mathbf c-\mathbf a\). Let \(X\) be the midpoint of \(OB\), so \(\overrightarrow{OX}=\tfrac12(\mathbf a+\mathbf c)\). Now find the midpoint of \(AC\) using position vectors: \(\tfrac12(\overrightarrow{OA}+\overrightarrow{OC})=\tfrac12(\mathbf a+\mathbf c)\). The two midpoints have the same position vector, so they are the same point. Therefore the diagonals meet at their common midpoint, which means they bisect each other. Notice how the proof uses only the rule for a midpoint and the fact that \(\overrightarrow{OB}=\mathbf a+\mathbf c\) in a parallelogram.

Worked example 16: Reading information from a diagram

In the diagram \(\overrightarrow{PQ}=2\mathbf u+\mathbf v\) and \(\overrightarrow{QR}=-\mathbf u+3\mathbf v\). Find \(\overrightarrow{PR}\) and \(\overrightarrow{RP}\), and say whether \(\overrightarrow{PR}\) is parallel to \(5\mathbf u+8\mathbf v\).

Add the two stages of the journey: \(\overrightarrow{PR}=\overrightarrow{PQ}+\overrightarrow{QR}=2\mathbf u+\mathbf v-\mathbf u+3\mathbf v=\mathbf u+4\mathbf v\). The reverse journey is the negative, so \(\overrightarrow{RP}=-\mathbf u-4\mathbf v\). For parallel vectors, one must be a multiple of the other. If \(5\mathbf u+8\mathbf v=k(\mathbf u+4\mathbf v)\) then \(k=5\) from the \(\mathbf u\) part but \(k=2\) from the \(\mathbf v\) part. These differ, so the vectors are not parallel.

Note: three ways to write a vector

A vector can be a column such as \(\begin{pmatrix}3\\-2\end{pmatrix}\), a bold letter such as \(\mathbf a\) (underlined \(\underline{a}\) when handwritten) or a pair of points with an arrow such as \(\overrightarrow{AB}\). You should move freely between the three. If a diagram has a grid, read column vectors from it by counting squares across and up. If the diagram has no grid, use letters and the rules for adding and subtracting. In exams the word magnitude means length, and "show that" means you must write every step of the argument, not only the result.

Method: a vector proof in four steps

Mark every given vector on the diagram and label unknown vectors with letters such as \(\mathbf a\) and \(\mathbf b\).
Find a route from the start point to the end point that uses only known vectors. Going backwards along a vector gives its negative.
Simplify by collecting the \(\mathbf a\) terms and the \(\mathbf b\) terms.
To show points are collinear, show that one vector is a multiple of the other and that they share a common point. Write a final sentence in words.

Watch out: the most common vector errors

Writing \(\overrightarrow{AB}=\mathbf a-\mathbf b\). It should be \(\mathbf b-\mathbf a\) because you go from \(A\) to \(B\) (end minus start).
Treating a column vector as a coordinate pair when adding: a vector is a movement, not a location, unless it is a position vector.
Showing only that two vectors are parallel when the question asks you to show that three points are in a straight line. You must also say that they share a point.
Dividing a line in a ratio with the wrong fraction: \(AP:PB=2:3\) means \(AP=\tfrac25AB\), not \(\tfrac23AB\).

Did you know?

The word vector comes from the Latin vehere, meaning "to carry". Vectors were developed in the 1880s by the American physicist Josiah Willard Gibbs and the English engineer Oliver Heaviside to describe forces and electric fields. Today the same ideas let a video game move thousands of characters across a screen and let a satellite navigation system add your speed to the speed of the wind.

Otto’s tip

To find a vector between two points, "go via the origin or a point you know". Write the route as a sum, for example \(\overrightarrow{AB}=\overrightarrow{AO}+\overrightarrow{OB}\), and simplify.

How to use this Year 11 vectors revision notes

First, read the key points so that you know where the Year 11 vectors revision notes is going. Then work through each definition and worked example, and cover the solution before you read it. After that, practise with the vectors worksheet and check every question with the worksheet answers. When you feel ready, test yourself with the multiple choice questions or the quick quiz. Finally, sit the vectors test and mark it with the test answers.

Also, the starters are a good warm-up, so you can keep the key facts fresh. Moreover, all the pages of this chapter work together, so you can move between them at any time. For example, many pupils read the lesson on Monday, practise on Wednesday and test themselves on Friday. In short, this Year 11 vectors revision notes gives you everything you need for the chapter.