Collinearity of Points: Solving the Equation

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The discussion focuses on proving the collinearity of points \(\alpha\), \(\beta\), and \(\gamma\) using the equation \(\lambda a + \mu b + \nu c = 0\). The key condition for collinearity is established as \(\frac{\lambda}{\alpha} + \frac{\mu}{\beta} + \frac{\nu}{\gamma} = 0\). Participants highlight the importance of the cross product equation \((p - q) \times (q - r) = 0\) in solving the problem. One user initially struggled with their solution but ultimately resolved it by recalling the cross product method. The discussion emphasizes the significance of using the correct equations to demonstrate collinearity.
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Homework Statement


The greek letters look like they're superscripted, they're not supposed to be.
a, b, c are vectors

given that
\lambdaa + \mu b + \nuc=0

show that the points \alphaa, \betab and \gammac are collinear if

\lambda/\alpha + \mu/\beta + \nu/\gamma = 0


Homework Equations



There are a lot of potentially relevant equations. Most important:
lines are collinear if a = xb

The Attempt at a Solution


My attempt is really long so I won't post it here, I'll just outline my method.

I found the line between \alphaa and \betab and said it was equal to x times the line between \betab and \gammac.

I also found a in terms of b and c from
\lambdaa +\mub + \nuc=0

and subbed it into the former equation. However I got stuck because I had an x that I couldn't get rid of.
 
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Hi thepopasmurf! :smile:
thepopasmurf said:
… There are a lot of potentially relevant equations. Most important:
lines are collinear if a = xb

Nooo … most important is the cross product equation, (p - q) x (q - r) = 0. :wink:
 
Thank you, solved it. I forgot about that one
 

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