Vectors

Dec 2009
755
7


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 ^^

thanks in advance
 
Jun 2009
806
275
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)
 
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Dec 2009
755
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thanks! mind if i ask u for part ii how do i handle the fraction?
 

Grandad

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Dec 2008
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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 \(\displaystyle (6,6)\)?

Grandad
 
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Dec 2009
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HEY Grandad !!! Many THX !!!!!!!!!!! your workings are excellent as usual, i fully understood them and have never heard about this method before!!!!!!!!!!!!!!!!