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VCE Stuff => VCE Mathematics => VCE Mathematics/Science/Technology => VCE Subjects + Help => VCE Specialist Mathematics => Topic started by: dave101 on February 04, 2016, 05:58:50 pm

Title: Vectors Help!He
Post by: dave101 on February 04, 2016, 05:58:50 pm: Hey guys,
I'm stuck on this vectors question from the new Cambridge spesh textbook from chapter 2 question 10 (a)

The question is .
ABCD is a trapezium with AB parallel to DC. X and Y are the midpoints of AD and BC respectively. (a) Express →XY in terms of a and b, where a = →AB and b = →DC.
Thanks in advance
:) ;) :D
Title: Re: Vectors Help!He
Post by: wyzard on February 05, 2016, 12:35:55 pm: This is a tricky one, the trick here is to draw another line that is parallel to AD and shift it to the right, join up with C to create a triangle on the right side. See the attached image.

Now from the trapezium we see that (I'm dropping off the arrow signs for convenience) $XY = AY - AX$ and since also $AY = AB + BY$ , this means that $XY = AB + BY - AX$ .

As X and Y is the midpoint of AD and BC respectively, we can say that $AX\quad =\quad \frac { 1 }{ 2 } AD$ and $BY\quad =\quad \frac { 1 }{ 2 } BC$ , so $XY\quad =\quad AB\quad +\quad \frac { 1 }{ 2 } BC\quad -\quad \frac { 1 }{ 2 } AD$ .

So what is $BC - AD$ ? Now referring back to the trapezium, remember I said the trick was to draw another parallel line to AD to form a triangle on the right? $BC-AD$ then is just simply the vector $BZ$ , which is just $b-a$ because of its direction.

From here just substitute back into (replacing $AB$ with $a$ ) $XY\quad =\quad a\quad +\quad \frac { 1 }{ 2 } (b-a)$ , obtaining $XY\quad =\quad \frac { 1 }{ 2 } (a+b)$
Title: Re: Vectors Help!He
Post by: dave101 on February 05, 2016, 05:17:42 pm: Wow! i would of never been able to think of that hahah.
Cheers, you are a true wyzard ;)
Title: Re: Vectors Help!He
Post by: wyzard on February 05, 2016, 10:19:07 pm: Quote from: dave101 on February 05, 2016, 05:17:42 pm
Wow! i would of never been able to think of that hahah.
Cheers, you are a true wyzard ;)
Yeah took me a while to figure that out :P