Vectors and translation vectorsIB MYP Maths Extended: Revision notes

A vector has a size and a direction. In two dimensions we write it as a column vector $\begin{pmatrix}x\\ y\end{pmatrix}$: $x$ units across (right is positive) and $y$ units up (up is positive). The vector from point $A$ to point $B$ is written $\overrightarrow{AB}$, and $\overrightarrow{AB}=B-A.$ For $A(1,1)$ and $B(6,2)$: $\overrightarrow{AB}=\begin{pmatrix}5\\ 1\end{pmatrix}$. Vectors with the same components are equal, wherever they are drawn. The vector $\overrightarrow{BA}$ goes the opposite way, so $\overrightarrow{BA}=-\overrightarrow{AB}=\begin{pmatrix}-5\\ -1\end{pmatrix}$. In print a vector may be written in bold, such as $\mathbf a$.

Work on each component separately.

• Add: $\begin{pmatrix}a\\ b\end{pmatrix}+\begin{pmatrix}c\\ d\end{pmatrix}=\begin{pmatrix}a+c\\ b+d\end{pmatrix}$.
• Subtract: $\begin{pmatrix}a\\ b\end{pmatrix}-\begin{pmatrix}c\\ d\end{pmatrix}=\begin{pmatrix}a-c\\ b-d\end{pmatrix}$.
• Multiply by a number (scalar) $k$: $k\begin{pmatrix}a\\ b\end{pmatrix}=\begin{pmatrix}ka\\ kb\end{pmatrix}$. Worked example. $\mathbf p=\begin{pmatrix}3\\ -2\end{pmatrix}$ and $\mathbf q=\begin{pmatrix}-1\\ 4\end{pmatrix}$. Then $\mathbf p+2\mathbf q=\begin{pmatrix}3\\ -2\end{pmatrix}+\begin{pmatrix}-2\\ 8\end{pmatrix}=\begin{pmatrix}1\\ 6\end{pmatrix}$. Multiplying by a negative number reverses the direction. Two vectors are parallel if one is a scalar multiple of the other.

The magnitude (length) of $\begin{pmatrix}x\\ y\end{pmatrix}$ comes from Pythagoras' theorem: $\left|\begin{pmatrix}x\\ y\end{pmatrix}\right|=\sqrt{x^2+y^2}.$ For $\mathbf p=\begin{pmatrix}3\\ -2\end{pmatrix}$, $|\mathbf p|=\sqrt{9+4}=\sqrt{13}$. The distance between points $A$ and $B$ is $|\overrightarrow{AB}|$. Magnitude is never negative, and the sign of a component does not matter because it is squared. Leave answers as surds when the question says exact, or round to 3 significant figures.

A translation moves every point of a shape the same distance in the same direction. It is described by a column vector: $\begin{pmatrix}3\\ -2\end{pmatrix}$ means 3 right and 2 down. The shape does not turn or change size.

• Image of a point $=$ object $+$ vector. $(2,5)$ translated by $\begin{pmatrix}3\\ -2\end{pmatrix}$ is $(5,3)$.
• The vector from an object point to its image is image $-$ object.
• The reverse translation is $-\mathbf v$.
• Two translations in a row add: $\begin{pmatrix}4\\ 3\end{pmatrix}$ then $\begin{pmatrix}-7\\ 2\end{pmatrix}$ is the single translation $\begin{pmatrix}-3\\ 5\end{pmatrix}$.

Vectors can be added along a path: $\overrightarrow{AC}=\overrightarrow{AB}+\overrightarrow{BC}$. Useful facts:

• In a parallelogram $ABCD$, $\overrightarrow{AD}=\overrightarrow{BC}$ and $\overrightarrow{AB}=\overrightarrow{DC}$.
• The position vector of the midpoint $M$ of $[AB]$ is $\overrightarrow{OM}=\frac12(\overrightarrow{OA}+\overrightarrow{OB})$.
• Parallel lines have parallel vectors. Worked example. Parallelogram $ABCD$ with $A(1,1)$, $B(6,2)$, $C(9,6)$. $\overrightarrow{BC}=\begin{pmatrix}3\\ 4\end{pmatrix}$, so $D=A+\begin{pmatrix}3\\ 4\end{pmatrix}=(4,5)$. The diagonal $\overrightarrow{AC}=\begin{pmatrix}8\\ 5\end{pmatrix}$ has length $\sqrt{89}$.