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VCE Stuff => VCE Mathematics => VCE Mathematics/Science/Technology => VCE Subjects + Help => VCE Specialist Mathematics => Topic started by: Blair1994 on April 01, 2011, 08:50:32 pm

Title: How do you know if points are collinear
Post by: Blair1994 on April 01, 2011, 08:50:32 pm: e.g (2,1) , (1, -3) and (3, 5)
Title: Re: How do you know if points are collinear
Post by: m@tty on April 01, 2011, 08:54:56 pm: "collinear" means lying on the same line. So construct a line that passes through two of the points and see if the third satisfies the relationship.
Title: Re: How do you know if points are collinear
Post by: xZero on April 01, 2011, 08:56:23 pm: make a line from (1,-3) to (3,5)

u = (1,-3) +t((3,5)-(1,-3))
u = (1,-3) +t(2,8)

x(t)=1+2t
y(t)=-3+8t

if (2,1) lines on this line, then the points are collinear, so sub the points in and see if t is the same for both parametric equations
Title: Re: How do you know if points are collinear
Post by: Blair1994 on April 02, 2011, 11:41:03 am: with a question like that where it asks about the relationship between points, is the answer usually collinear? or should i test other stuff?
Title: Re: How do you know if points are collinear
Post by: m@tty on April 02, 2011, 12:17:47 pm: Ah, you can also have co-planar points if they give you ordered triples [ (x,y,z) ] but I can't remember if that was in spesh or not..

But you need to approach each question independent of previous ones you've done.. after you deduce what it is asking for, formulate a method to tackle it. Don't use stock standard approaches for similar questions.

Quote from: xZero on April 01, 2011, 08:56:23 pm
make a line from (1,-3) to (3,5)

u = (1,-3) +t((3,5)-(1,-3))
u = (1,-3) +t(2,8)

x(t)=1+2t
y(t)=-3+8

if (2,1) lines on this line, then the points are collinear, so sub the points in and see if t is the same for both parametric equations

I would just use normal cartesian equations - easier. (I still find it simpler, even though the working here is longer haha)

so, using (2,1) and (1,-3):

$m=\frac{y_2-y_1}{x_2-x_1}=\frac{1-(-3)}{2-1}=4$

then sub one of the points into
$y-y_1=m(x-x_1)$

sub (2,1)
$y-1=4(x-2) \iff y=4x-7$

Try the third point(3,5):

$\text{LHS}=5$

$\text{RHS}=4(3)-7=5$

$\therefore \text{LHS}=\text{RHS}$

And so the points are ~~not~~ collinear.
Title: Re: How do you know if points are collinear
Post by: Blair1994 on April 02, 2011, 12:20:41 pm: the other point is 3,5 :P
Title: Re: How do you know if points are collinear
Post by: xZero on April 02, 2011, 12:24:46 pm: Haha I always prefer parametric equations, also you subbed the points wrong. It should be (3,5) and they are collinear
Title: Re: How do you know if points are collinear
Post by: m@tty on April 02, 2011, 12:52:05 pm: Quote from: Blair1994 on April 02, 2011, 12:20:41 pm
the other point is 3,5 :P

Quote from: xZero on April 02, 2011, 12:24:46 pm
Haha I always prefer parametric equations, also you subbed the points wrong. It should be (3,5) and they are collinear

Indeed xD I'm at work, let's say it was rushed..

And you prefer parametric? Fair enough ;)

btw, you're missing a t in your parametric equation for y... :P
Title: Re: How do you know if points are collinear
Post by: vea on April 02, 2011, 06:20:04 pm: There is also the vector method:

If any two vectors from one point to another are parallel, then the three points are collinear.
e.g (2,1) , (1, -3) and (3, 5)

(2,1)-(1,-3)=(1,4)
(2,1)-(3,5)=(-1,-4)

Now (2,1)-(3,5)=-[(2,1)-(1,-3)]
Therefore the points (2,1), (1,-3) and (3,5) are collinear.

I don't think I've done it formally here but to do it properly, you would assign the points above vectors and show it using vectors.