Vector Arithmetic and Representations

Vectors describe a movement

A vector records how far and in which direction something moves. In two dimensions, it can be written as a column vector:

$$\begin{pmatrix}x\\y\end{pmatrix}.$$

The top component, $x$, is the horizontal movement: positive means right and negative means left. The bottom component, $y$, is the vertical movement: positive means up and negative means down. Thus $\begin{pmatrix}4\\-3\end{pmatrix}$ means 4 units right and 3 units down.

A vector can also be drawn as an arrow. Its direction shows the direction of movement and its length represents the magnitude. Moving the whole arrow to another position without rotating or resizing it does not change the vector: only its magnitude and direction matter.

Adding vectors

Adding vectors combines movements performed one after another. Suppose

$$\mathbf a=\begin{pmatrix}a_x\\a_y\end{pmatrix} \quad\text{and}\quad \mathbf b=\begin{pmatrix}b_x\\b_y\end{pmatrix}.$$

Add corresponding components:

$$\mathbf a+\mathbf b= \begin{pmatrix}a_x+b_x\\a_y+b_y\end{pmatrix}.$$

For example,

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

Horizontally, 5 units right followed by 3 units left leaves 2 units right. Vertically, the movement remains 2 units down.

Diagrammatically, place the tail of the second vector at the head of the first. The resultant runs from the original starting point to the final endpoint. The order can be reversed without changing the resultant: $\mathbf a+\mathbf b=\mathbf b+\mathbf a$.

Subtracting vectors

Subtract corresponding components in the same way:

$$\mathbf a-\mathbf b= \begin{pmatrix}a_x-b_x\\a_y-b_y\end{pmatrix}.$$

Subtraction can also be understood as adding the opposite vector:

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

For instance,

$$\begin{pmatrix}4\\1\end{pmatrix}- \begin{pmatrix}-2\\3\end{pmatrix}= \begin{pmatrix}4-(-2)\\1-3\end{pmatrix}= \begin{pmatrix}6\\-2\end{pmatrix}.$$

The double negative in the top component is important. On a diagram, $-\mathbf b$ has the same magnitude as $\mathbf b$ but points in the opposite direction. Draw $\mathbf a$ and then $-\mathbf b$ head to tail to obtain $\mathbf{a}$.

Multiplying by a scalar

A scalar is an ordinary number. To multiply a vector by a scalar $k$, multiply every component by $k$:

$$k\begin{pmatrix}a_x\\a_y\end{pmatrix}= \begin{pmatrix}ka_x\\ka_y\end{pmatrix}.$$

If $\mathbf a=\begin{pmatrix}-2\\5\end{pmatrix}$, then

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

The value of $k$ determines the diagrammatic effect:

• if $k>1$, the vector keeps its direction and becomes $k$ times as long;
• if $0<k<1$, it keeps its direction but becomes shorter;
• if $k=1$, it is unchanged;

In particular, $-\mathbf a$ has the same magnitude as $\mathbf a$ but the opposite direction. Multiplying by $-2$ reverses the vector and doubles its magnitude.

Combining operations

Vector arithmetic follows the usual priority of operations. Scalar multiplication is completed before addition or subtraction unless brackets say otherwise. For example, if

$$\mathbf a=\begin{pmatrix}1\\4\end{pmatrix} \quad\text{and}\quad \mathbf c=\begin{pmatrix}-1\\-2\end{pmatrix},$$

then

$$3\mathbf a-2\mathbf c =\begin{pmatrix}3\\12\end{pmatrix}- \begin{pmatrix}-2\\-4\end{pmatrix} =\begin{pmatrix}5\\16\end{pmatrix}.$$

A complicated expression is safest when each scalar multiple is calculated first and the resulting columns are then combined row by row. The final column vector is the resultant: the one movement with the same overall effect as the entire expression.