Vectors in two dimensionsAQA A-Level Maths: Revision notes

Section 1

Vectors and how we write them

A vector has both magnitude (size) and direction; a scalar has size only. In print a vector is written in bold, $\mathbf a$ , or as $\overrightarrow{AB}$ , the vector from $A$ to $B$ ; by hand underline it, $\underline{a}$ . In two dimensions a vector can be written as a column vector $\begin{pmatrix}x\\ y\end{pmatrix}$ or in terms of the unit vectors $\mathbf i$ and $\mathbf j$ as $x\mathbf i+y\mathbf j$ , where $\mathbf i$ is one unit along the $x$ -axis and $\mathbf j$ one unit along the $y$ -axis. The numbers $x$ and $y$ are the components.

Key termsvectorscalarunit vectorscomponents

Common mistake

Writing a vector with no arrow or bold, so that it looks like a number. Two vectors are equal only if they have the same direction and size, wherever they are drawn.

Section 2

Adding and subtracting vectors

Algebraically, add or subtract component by component: $\begin{pmatrix}a_1\\ a_2\end{pmatrix}+\begin{pmatrix}b_1\\ b_2\end{pmatrix}=\begin{pmatrix}a_1+b_1\\ a_2+b_2\end{pmatrix}$ . Geometrically, the triangle law: to add $\mathbf a$ and $\mathbf b$ , draw $\mathbf b$ starting where $\mathbf a$ ends; the sum is the vector from the start of $\mathbf a$ to the end of $\mathbf b$ , written $\overrightarrow{AB}+\overrightarrow{BC}=\overrightarrow{AC}$ . In a parallelogram with sides $\mathbf a$ and $\mathbf b$ from one corner, the diagonal from that corner is $\mathbf a+\mathbf b$ and the other diagonal is $\mathbf b-\mathbf a$ . Subtraction is addition of the negative: $\mathbf a-\mathbf b=\mathbf a+(-\mathbf b)$ , and $\overrightarrow{BA}=-\overrightarrow{AB}$ .

Key termstriangle lawresultant

Exam tip

To find any vector, pick a route along known vectors from the start to the end, and reverse the sign of any vector you travel against.

Section 3

Multiplying by a scalar

Multiplying by a scalar $k$ multiplies every component: $k\begin{pmatrix}x\\ y\end{pmatrix}=\begin{pmatrix}kx\\ ky\end{pmatrix}$ . Geometrically the new vector is $|k|$ times as long, in the same direction if $k>0$ and the opposite direction if $k<0$ . So $-\mathbf a$ is the same length as $\mathbf a$ but reversed, and $\frac12\overrightarrow{AB}$ points from $A$ halfway to $B$ . Ratio statements translate directly: if $PX:XR=2:1$ with $X$ on $PR$ , then $\overrightarrow{PX}=\frac23\overrightarrow{PR}$ .

Key termsscalar multiple

Common mistake

Writing $\overrightarrow{PX}=\frac12\overrightarrow{PR}$ for $PX:XR=2:1$ . The ratio $2:1$ makes $PX$ two of three equal parts, so $\frac23$ .

Section 4

Parallel vectors

Two non-zero vectors are parallel if one is a scalar multiple of the other: $\mathbf v=k\mathbf u$ . In components the ratios must match: $2\mathbf i+5\mathbf j$ and $\lambda\mathbf i-4\mathbf j$ are parallel when $\lambda=2k$ and $-4=5k$ , so $k=-\frac45$ and $\lambda=-\frac85$ . Parallel vectors need not point the same way: a negative $k$ means opposite directions. A vector parallel to $\mathbf i$ has zero $\mathbf j$ -component, and one parallel to $\mathbf j$ has zero $\mathbf i$ -component.

Key termsparallel vectors

Exam tip

Equate components to find unknowns: $x_1\mathbf i+y_1\mathbf j=x_2\mathbf i+y_2\mathbf j$ means $x_1=x_2$ and $y_1=y_2$ .

Section 5

Worked example: a parallelogram

$ABCD$ is a parallelogram with $\overrightarrow{AB}=\mathbf a$ and $\overrightarrow{AD}=\mathbf b$ . Opposite sides are equal and parallel, so $\overrightarrow{BC}=\mathbf b$ and $\overrightarrow{DC}=\mathbf a$ .

$\overrightarrow{AC}=\overrightarrow{AB}+\overrightarrow{BC}=\mathbf a+\mathbf b$ .
$\overrightarrow{BD}=\overrightarrow{BA}+\overrightarrow{AD}=-\mathbf a+\mathbf b$ .
If $M$ is the midpoint of $DC$ : $\overrightarrow{AM}=\overrightarrow{AD}+\frac12\overrightarrow{DC}=\mathbf b+\frac12\mathbf a$ .

Exam tip

Label equal vectors on a sketch (for example both $\overrightarrow{AD}$ and $\overrightarrow{BC}$ as $\mathbf b$ ) before you start.

That's the notes covered.

Carry on to the next subtopic.

Exam questions on Vectors in two dimensions

The vectors $\mathbf a$ and $\mathbf b$ are given by $\mathbf a=\begin{pmatrix}3\\ -2\end{pmatrix}$ and $\mathbf b=\begin{pmatrix}-1\\ 4\end{pmatrix}$ .
Find the value of $p$ for which $\mathbf a+p\mathbf b$ is parallel to the vector $\mathbf i$ .2 marks
$ABCD$ is a parallelogram with $\overrightarrow{AB}=\mathbf a$ and $\overrightarrow{AD}=\mathbf b$ .
The point $M$ is the midpoint of $DC$ . Find $\overrightarrow{AM}$ in terms of $\mathbf a$ and $\mathbf b$ .2 marks
$\mathbf u=2\mathbf i+5\mathbf j$ , $\mathbf v=\lambda\mathbf i-4\mathbf j$ and $\mathbf w=\mathbf i-\mathbf j$ , where $\lambda$ is a constant.
Given that $\mathbf u$ and $\mathbf v$ are parallel, find the value of $\lambda$ .3 marks

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Written by the Exaim team, led by Shaun Daswani (Head of Upper Secondary, Improve ME Institute; MSc Financial Mathematics, Imperial College London; BSc, UCL) and Jason Daswani (operational lead, Improve ME Institute; LSE).