# Collinearity of Points: Solving the Equation

• thepopasmurf
In summary, the conversation discusses the collinearity of points \alphaa, \betab, and \gammac and how it relates to the equation \lambdaa + \mu b + \nuc=0. The important equation for determining collinearity is the cross product equation, (p - q) x (q - r) = 0.
thepopasmurf

## Homework Statement

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

given that
$$\lambda$$a + $$\mu$$ b + $$\nu$$c=0

show that the points $$\alpha$$a, $$\beta$$b and $$\gamma$$c 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 $$\alpha$$a and $$\beta$$b and said it was equal to x times the line between $$\beta$$b and $$\gamma$$c.

I also found a in terms of b and c from
$$\lambda$$a +$$\mu$$b + $$\nu$$c=0

and subbed it into the former equation. However I got stuck because I had an x that I couldn't get rid of.

Hi thepopasmurf!
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.

Thank you, solved it. I forgot about that one

## 1. What is collinearity of points?

Collinearity of points refers to a geometric concept where three or more points lie on the same straight line.

## 2. How is the collinearity of points determined?

The collinearity of points can be determined by solving the equation of the line that passes through the given points. If the equation is satisfied by all the points, then they are collinear.

## 3. What is the equation used to determine collinearity of points?

The equation used is the slope-intercept form of a line, y = mx + b, where m is the slope and b is the y-intercept. By plugging in the coordinates of the points, we can determine if they satisfy the equation.

## 4. Can more than three points be collinear?

Yes, collinearity can involve any number of points as long as they all lie on the same straight line.

## 5. How is collinearity of points useful in geometry?

Collinearity of points is useful in geometry for determining the orientation of points in a plane, as well as for solving problems involving intersecting lines and finding the equation of a line passing through given points.

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