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 September 10th, 2012, 09:31 PM #1 Senior Member   Joined: Apr 2010 Posts: 128 Thanks: 0 Scalar vectors : need help in proving In my exercise book got this question under topic Vectors Geometry -->Scalar vectors $\text The three points A \ , B \ , C \ , \text have position vectors a,b and c respectively. If c = 3b-2a. Show that the points A , B and C are collinear$ i have absolutely no idea how to do it. I even doubt the validity of this question. All I can imagine is that there are 3 lines flying around without directions(i imagined in 2d)
 September 10th, 2012, 09:50 PM #2 Senior Member     Joined: Jul 2010 From: St. Augustine, FL., U.S.A.'s oldest city Posts: 12,211 Thanks: 521 Math Focus: Calculus/ODEs Re: Scalar vectors : need help in proving We may give the coordinates of A as: $$$x_a,y_a$$$, B as $$$x_b,y_b$$$ and C as $$$3x_b-2x_a,3y_b-2y_a$$$. These points are collinear if the slope from A to B is equal to that from B to C: $\frac{y_b-y_a}{x_b-a_a}=\frac{3y_b-2y_a-y_b}{3x_b-2x_a-x_b}=\frac{2y_b-2y_a}{2x_b-2x_b}=\frac{y_b-y_a}{x_b-a_a}$ Hence, the three points are collinear.
 September 13th, 2012, 09:23 PM #3 Senior Member   Joined: Apr 2010 Posts: 128 Thanks: 0 Re: Scalar vectors : need help in proving it took me days.....finally sort out.... $AB= b-a$ $AC= c-a$ $OC= 3b-2a$ $AC= 3b-2a-a$ $AC= 3(AB)$ therefore they are collinear....... but how come i don't understand your equation, mark? i mean , how you get the 4 '=' sign line up to each other and say it is collinear.....
 September 13th, 2012, 09:26 PM #4 Senior Member     Joined: Jul 2010 From: St. Augustine, FL., U.S.A.'s oldest city Posts: 12,211 Thanks: 521 Math Focus: Calculus/ODEs Re: Scalar vectors : need help in proving If the slope between the first point to the second point is the same as the slope from the second point to the third point, then we know the three points are collinear.
September 14th, 2012, 07:25 AM   #5
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Re: Scalar vectors : need help in proving

Hello, rnck!

I'll explain what MarkFL did . . .

Quote:
 $\text {The three points }A,\,B,\,C\;\text{ have position vectors }\vec a,\,\vec b\;\text{ and }\vec c\text{, respectively.} \tex{If } \vec c \:=\: 3\vec b\,-\,2\vec a,\;\text{ show that the points }A,\,B\;\text{ and }C\;\text{ are collinear.}$

$\text{The three points are: }\:A(x_a,\,y_a),\;B(x_b,\,y_b),\;C(3x_b-2x_a,\:3y_b-2y_a)$

$\text{The slope of }AB\text{ is: }\:m_{_{AB}} \;=\;\frac{y_b\,-\,y_a}{x_b\,-\,x_a}$

$\text{The slope of }BC\text{ is: }\:m_{_{BC}}\;=\;\frac{(3y_b-2y_a)\,-\,y_b}{(3x_b-2x_a)\,-\,x_b} \;=\;\frac{2y_b-2y_a}{2x_b-2x_a} \;=\;\frac{2(y_b-y_a)}{2(x_b-x_a)} \;=\; \frac{y_b\,-\,y_a}{x_b\,-\,x_a}$

$\text{Since }AB\text{ and }BC\text{ have the same slope, and they share point }B,
\;\;\;A,\,B\text{ and }C\text{ are collinear.}$

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