Troubleshooting Vector Equations: Proving Collinearity of 3 Concurrent Lines

[Karate Chop]

Hi, I'm having a few troubles with this question on my assignment. I've tried for many hours to get out the answer but i keep getting stuck and am not sure if I'm going about it the right way.

This is the question:

Three concurrent lines OA, OB and OC are produced to D, E and F respectively. Prove, using vectors, that the point of intersection of AB and DE, BC and EF, CA and FD are collinear.

What worked out the equations of the lines going through the points A and B, D and E, etc. and then made the equations of lines through A and B and D and E equal each other, since they were the position vectors of any point along that line. I did this inorder to try to get the points D, E and F in terms of a, b and c (the position vectors of A, B and C respectively), however in my last and most successful attempt at this question, i end up with 3 sets of two simultaneous equations, each set had three different variables so i couldn't solve it. Please help! thanks in advance. john.

 

[misogynisticfeminist]

don't you think its better if you gave us the equation of the position vectors?

 

[tacman]

Hi John,

It seems like you're on the right track by setting up the equations of the lines and trying to find a way to express the points D, E, and F in terms of the position vectors A, B, and C. However, instead of trying to solve for the points directly, you can try to prove that the points are collinear using the properties of vectors.

First, recall the definition of collinearity - three points are collinear if they lie on the same line. In this case, we can show that the points of intersection AB and DE, BC and EF, and CA and FD all lie on the same line by proving that their corresponding position vectors are parallel.

To do this, we can use the fact that the cross product of two parallel vectors is equal to the zero vector. So, for example, if we take the cross product of the vectors AB and DE, we should get the zero vector if they are parallel.

If you're not familiar with the cross product, it is defined as follows: given two vectors u and v, the cross product u x v is a vector that is perpendicular to both u and v, and has a magnitude equal to the product of the magnitudes of u and v multiplied by the sine of the angle between them.

So, in this case, we can calculate the cross product AB x DE and if it is equal to the zero vector, then we know that AB and DE are parallel. If we repeat this process for the other pairs of intersecting lines, we should get the same result - all of the cross products should be equal to the zero vector.

If you're still having trouble, it might be helpful to draw a diagram and label all of the vectors involved. This can help visualize the problem and make it easier to see the relationships between the vectors.

Hope this helps! Good luck with your assignment.