# Vectors

#### Punch

sry i but i really have no idea how to start the question... i have problems with both parts of the question as they are similar...

i am not asking u to spoonfeed me, perhaps u can solve the first part and let me solve the second part as they are similar ^^

#### sa-ri-ga-ma

Shift y from the present position to (2,1) and (4,3).

Complete the parallelogram. The resultant of x and y will be (1,4) and (4,3)

Parallel to this resultant from P is (0,5) and (3,4)

Punch

#### Punch

thanks! mind if i ask u for part ii how do i handle the fraction?

#### sa-ri-ga-ma

thanks! mind if i ask u for part ii how do i handle the fraction?
3y/2 will be (4, 1) and (7, 4)

Shift x to the right so that x touches y.

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MHF Hall of Honor
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 $$\displaystyle \textbf{x} + \textbf{y}$$ means
Move 1 unit to the right and 3 units down
and then
Move 2 units to the right and 2 units up
which is obviously equivalent to
Move 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 $$\displaystyle (6,6)$$?