Vector Arithmetic and Representations

Vectors are equal when they have the same magnitude and direction, even if their arrows start in different places. You can therefore slide an arrow to a new starting point without changing the vector, provided you do not rotate it or change its length.

Adding vectors joins movements head to tail

Adding vectors means making one movement followed by another. Draw the first arrow, then place the tail of the second arrow at the head of the first. The resultant vector joins the original starting point directly to the final finishing point.

For three points A, B and C, travelling from A to B and then B to C gives the same overall movement as travelling directly from A to C:

$$\overrightarrow{AB}+\overrightarrow{BC}=\overrightarrow{AC}.$$

Column notation records the same process by adding corresponding components. Suppose $\mathbf{a}=\begin{pmatrix}5\\-2\end{pmatrix}$ and $\mathbf{b}=\begin{pmatrix}-3\\0\end{pmatrix}$. The first movement is 5 right and 2 down; the second is 3 left with no vertical movement. Therefore:

$$\mathbf{a}+\mathbf{b}=\begin{pmatrix}5+(-3)\\-2+0\end{pmatrix}=\begin{pmatrix}2\\-2\end{pmatrix}.$$

The overall movement is 2 right and 2 down. Reversing the order of these two movements reaches the same finishing point, so $\mathbf{a}+\mathbf{b}=\mathbf{b}+\mathbf{a}$.

Subtracting a vector reverses its movement

The vector $-\mathbf{b}$ has the same length as $\mathbf{b}$ but points in the opposite direction. Subtraction means adding this reversed vector:

$$\mathbf{a}-\mathbf{b}=\mathbf{a}+(-\mathbf{b}).$$

Using the same vectors, $\mathbf{b}$ is 3 left, so $-\mathbf{b}$ is 3 right. After making movement $\mathbf{a}$, move another 3 right:

$$\mathbf{a}-\mathbf{b}=\begin{pmatrix}5-(-3)\\-2-0\end{pmatrix}=\begin{pmatrix}8\\-2\end{pmatrix}.$$

Reversing a named arrow follows the same rule: $\overrightarrow{BA}=-\overrightarrow{AB}$. Unlike addition, subtraction depends on the order: is the reverse of .

Multiplying by a scalar changes the arrow's size and sometimes its direction

A scalar is a number with no direction. To multiply a vector by a scalar, multiply both components by that number. For example, if $\mathbf{a}=\begin{pmatrix}-2\\5\end{pmatrix}$, then:

$$3\mathbf{a}=\begin{pmatrix}3(-2)\\3(5)\end{pmatrix}=\begin{pmatrix}-6\\15\end{pmatrix}.$$

Both movements are tripled, so the arrow becomes three times as long while keeping its direction.

For a non-zero vector, multiplying by a positive number greater than 1 stretches the arrow; multiplying by a positive number between 0 and 1 shortens it. For instance, $\begin{pmatrix}1\\1\end{pmatrix}$ becomes $\begin{pmatrix}2\\2\end{pmatrix}$ when doubled, or when halved.

A negative multiplier also reverses the direction. Multiplication by $-2$ doubles the length and reverses the arrow; multiplication by $-1$ only reverses it. Multiplication by 0 gives $\begin{pmatrix}0\\0\end{pmatrix}$, the zero vector: there is no movement and no direction.

Finding a route through a shape

Vector arithmetic also works when vectors are given as letters rather than columns. In rectangle ABCD, let $\overrightarrow{AB}=\mathbf{x}$ and $\overrightarrow{AD}=\mathbf{y}$. Opposite sides have equal lengths, and the arrows from A to B and D to C point in the same direction, so .

To find $\overrightarrow{AC}$, take the route A to D to C:

$$\overrightarrow{AC}=\overrightarrow{AD}+\overrightarrow{DC}=\mathbf{y}+\mathbf{x}.$$

For $\overrightarrow{BD}$, take B to A to D. The first part goes against the arrow for $\mathbf{x}$:

$$\overrightarrow{BD}=\overrightarrow{BA}+\overrightarrow{AD}=-\mathbf{x}+\mathbf{y}.$$

If M is the midpoint of CD, the movement from D to M is half of the movement from D to C. Hence:

$$\overrightarrow{AM}=\overrightarrow{AD}+\overrightarrow{DM}=\mathbf{y}+\frac12\mathbf{x}.$$

The same route method works with longer expressions. If $\overrightarrow{AB}=5\mathbf{a}+4\mathbf{b}$ and $\overrightarrow{BC}=2\mathbf{a}-3\mathbf{b}$, then:

$$\overrightarrow{AC}=(5\mathbf{a}+4\mathbf{b})+(2\mathbf{a}-3\mathbf{b})=7\mathbf{a}+\mathbf{b}.$$

Collect the $\mathbf{a}$ terms together and the $\mathbf{b}$ terms together, just as you would collect like terms in algebra.

When two vectors start at the same point O, a route from A to B goes back to O and then out to B. Thus $\overrightarrow{AB}=\overrightarrow{OB}-\overrightarrow{OA}$. For example, if and :

$$\overrightarrow{AB}=(\mathbf{a}-2\mathbf{b})-(4\mathbf{a}-3\mathbf{b})=-3\mathbf{a}+\mathbf{b}.$$