Movement (displacement) vectors

Hello Punch

From the diagram, we can see that $\displaystyle \textbf{x}$ represents a movement which can be described as:Move 1 unit to the right and 3 units down.

We can write this as$\displaystyle \textbf{x}=\binom{1}{-3}$

Similarly $\displaystyle \text{y}$ represents the movement:Move 2 units to the right and 2 units up

or:$\displaystyle \textbf{y}=\binom{2}{2}$

We add two movement (or displacement) vectors together using an ordinary $\displaystyle +$ sign, by thinking of $\displaystyle +$ as representing the phrase 'and then'. So $\displaystyle \textbf{x} + \textbf{y}$ meansMove 1 unit to the right and 3 units downand then

Move 2 units to the right and 2 units up

which is obviously equivalent toMove 3 units to the right and 1 unit down

We sometimes write all of this as:$\displaystyle \textbf{x} + \textbf{y}=\binom{1}{-3} + \binom{2}{2}$$\displaystyle = \binom{3}{-1}$

Do you see how it works?

Now $\displaystyle \vec{PR}$ stands for:Move from $\displaystyle P$ to $\displaystyle R$

So if $\displaystyle \vec{PR}=\textbf{x} + \textbf{y}$ and we must start at the point $\displaystyle P\; (0,5)$, and carry out the movement $\displaystyle \binom{3}{-1}$. When we do this, we get to the point $\displaystyle (3,4)$. So that's where $\displaystyle R$ is: $\displaystyle (3,4)$.

Now $\displaystyle \frac32\textbf{y}$ is the same movement as $\displaystyle \textbf y$, but multiplied by $\displaystyle \frac32$. In other words:$\displaystyle \frac32\textbf{y}=\frac32\times\binom22$$\displaystyle =\binom33$

So:$\displaystyle \frac32\textbf{y}-\textbf x=\binom33 - \binom{1}{-3}$$\displaystyle =\binom{2}{6}$ (Check this carefully!)

and this is the movement $\displaystyle \vec{AB}$. So, to find the position of the point $\displaystyle B$, start at $\displaystyle A$ and carry out the movement $\displaystyle \binom26$. Can you see that $\displaystyle B$ is at $\displaystyle (6,6)$?

Grandad