Proving Non-Colinear Points in IR3

[esmeco]

Homework Statement

The problem states the following: Show that the points of space A(3,1,-5) B(8,3,3) and C(2,1,-4) are not colinear

Homework Equations

I've tried to use the equation y2-y1=x2-x1 for the straight line on IR2 but since we are working in IR3 the same formula doesn't apply

The Attempt at a Solution

I'm not sure how we calculate the slope of the straight line on IR3 or if there's another formula to demonstrate that the points are non colinear.

Thanks in advance for the reply!

 

[OlderDan]

You could create displacement vectors from one of the points to the other two and take the dot product or cross product of the two vectors. Either way, if you know how the dot and cross products of parallel vectors behave, you can reach the needed conclusion from either one of these products.

 

[esmeco]

...

Let me see if I understood this correctly...I should make 2 vectors out of these 3 points,lets say vector AB and vector BC.To demonstrate that the points are non-colinear, calculating the dot or cross product they must be different than zero.If vector AB and vector BC are not multiple of one another(hence not parallel), the vectors are non colinear,hence the points are non colinear.Am I right on this?Feel free to correct me if I'm wrong!Thanks in advance for the reply!

 

[OlderDan]

Let me see if I understood this correctly...I should make 2 vectors out of these 3 points,lets say vector AB and vector BC.To demonstrate that the points are non-colinear, calculating the dot or cross product they must be different than zero.If vector AB and vector BC are not multiple of one another(hence not parallel), the vectors are non colinear,hence the points are non colinear.Am I right on this?Feel free to correct me if I'm wrong!Thanks in advance for the reply!

"Different than zero" would come in if you chose to use the cross product to test for colinearity, but not if you use the dot product. What must be true about the cross product of parallel vectors? How does the dot product depend on the angle between two vectors?

 

[esmeco]

After searching a bit, I think I'm correct now(but again feel free to correct me if I'm wrong): The cross product of vectors must be zero,hence if it's zero their parallel,and since their parallel their colinear.Using the dot product, if the result of that product equals one, the vectors are parallel.
Am I right?

 

[OlderDan]

After searching a bit, I think I'm correct now(but again feel free to correct me if I'm wrong): The cross product of vectors must be zero,hence if it's zero their parallel,and since their parallel their colinear.Using the dot product, if the result of that product equals one, the vectors are parallel.
Am I right?

The dot product does not have to be one, but the ratio of the dot product to the product of the lengths of the individual vectors must be one if the points are colinear. The cross product will be zero if they are colinear. More generally these conditions apply to any pair of parallel vectors. They do not have to be displacement vectors formed by colinear points.

Another way to show that your points are colinear is to show that one vector is a multiple of the other. If you compare the ratios of corresponding components, all the ratios will be the same if the points are colinear.

 

[esmeco]

Thank you for your help!I now understand it!