22.1 Vector notation and column vectors — Practise with help

1. $(5,-1)$: add 3 to the horizontal coordinate and subtract 2 from the vertical coordinate.
2. $-1-4=-5$, so the change is 5 units left.

Add the top component to the horizontal coordinate and the lower component to the vertical coordinate.

The new coordinates are $(-2+4,\ 3-5)$.

$(2,-2)$, because $-2+4=2$ and $3-5=-2$.

Subtract the start coordinates from the end coordinates.

The changes are $-1-3$ horizontally and $4-(-2)$ vertically.

$\overrightarrow{AB}=\begin{pmatrix}-4\\6\end{pmatrix}$: move 4 left and 6 up.

Read the top entry as horizontal movement and the lower entry as vertical movement.

Both entries are negative, so both movements are in the negative directions.

Move 3 units left and 2 units down. This is a 2D translation vector.

Find end minus start for each coordinate.

Horizontal change $=2-(-1)$; vertical change $=0-2$.

$\begin{pmatrix}3\\-2\end{pmatrix}$.

Start at $(0,0)$, then move right before moving up.

Mark the endpoint $(2,3)$ and draw an arrow from the origin.

The arrow runs from $(0,0)$ to $(2,3)$. Any parallel arrow with the same direction and length represents the same vector.

Follow the route from $A$ to $B$, then from $B$ to $C$.

The route is $\mathbf{u}$ followed by $\mathbf{v}$.

$\overrightarrow{AC}=\mathbf{u}+\mathbf{v}$. We will calculate such sums in section 22.2.