Vectors

Vector Notation

There are several ways a vector can be represented, so it’s important you are familiar with them all to prepare for your exams.

Bold / Underlined

The most common way you’ll see is a letter written in bold, for example, these two lines are vectors:

For example, the vector \boldsymbol{b} is in the direction of the pink arrow.

Note: when you are writing in your exam, as you can’t write in bold, you will be expected to underline vectors, like: \underline{a}

Column Notation

Vectors may also be displayed in columns:

Example: The column vector \begin{pmatrix}2 \\ 1\end{pmatrix} means 2 spaces to the right and 1 space up.

If you had \begin{pmatrix}-2 \\ -1\end{pmatrix}, this would mean 2 spaces to the left and 1 space down.

Arrow

Finally, you may see vectors written like:

\overrightarrow{AB} which is the vector between point A and point B

\overrightarrow{OC} which is the vector between the origin and point C

Note: The point O almost always represents the origin in vector questions.

Adding and Subtracting Vectors

When adding and subtracting using diagrams, it is important to pay attention to the direction the vector arrow is going.

A vector \boldsymbol{a} is positive in one direction, and if you flip the direction (point the arrow towards the opposite direction), the vector is now \boldsymbol{-a}:

Adding

If we have two vectors, \boldsymbol{a} and \boldsymbol{b}, these can be added:

\boldsymbol{a}+\boldsymbol{b}

\boldsymbol{a}+\boldsymbol{b} will follow a route from the start of the \boldsymbol{a} arrow, to the end of the \boldsymbol{b} arrow, both in the positive direction:

Subtracting

For subtraction, for example,

\boldsymbol{a}-\boldsymbol{b}

The \boldsymbol{b} is negative, so either of these diagrams correctly portray this subtraction:

Adding and Subtracting Column Vectors

Column vectors are easy to add and subtract, because you just treat the top row and the bottom row as separate sums, for example:

\begin{pmatrix}5 \\ 2\end{pmatrix}+\begin{pmatrix}3 \\ 1\end{pmatrix}=\begin{pmatrix}5+3 \\ 2+1\end{pmatrix}=\begin{pmatrix}8 \\ 3\end{pmatrix}

\begin{pmatrix}6 \\ 4\end{pmatrix}-\begin{pmatrix}2 \\ 3\end{pmatrix}=\begin{pmatrix}6-2 \\ 4-3\end{pmatrix}=\begin{pmatrix}4 \\ 1\end{pmatrix}

Multiplying Vectors by a Scalar

Multiplying vectors by a scalar simply means multiplying vectors by a number (not a vector).

 

For example,

\boldsymbol{a}\times3=3\boldsymbol{a}

This is represented in the diagram:

If we have a vector containing more than one letter, we just multiply each letter by the scalar:

3\times(\boldsymbol{a}+\boldsymbol{b})=3\boldsymbol{a}+3\boldsymbol{b}

Similarly, in column vector form:

4\times\begin{pmatrix}3 \\ 2\end{pmatrix}=\begin{pmatrix}4\times3 \\ 4\times2\end{pmatrix}=\begin{pmatrix}12 \\ 8\end{pmatrix}

Vector Magnitudes

The magnitude of a vector is the length of a vector.

Magnitude has no direction, so is always positive.

Example:

Find the magnitude of the vector

\overrightarrow{XY}=\begin{pmatrix}3 \\ 4\end{pmatrix}

Looking at the diagram, we can work out the length of the vector \overrightarrow{XY} by treating it as a right-angled triangle and using Pythagoras’ theorem:

\sqrt{4^2+3^2}=5