Vectors???

I've got a few questions to do with vectors:





1) Let u, v and w be three non-collinear points vectors in IR^n, and suppose that





OC = -2u + 3v +5w


OD = u + 2v + 3w


OE = 7u - w





I have to show that C, D and E are collinear points. So to do this I showed that all three vectors are scalar multiples of each other. Which means they are all parallel which means they are collinear.


My question is: Is that all is required to show that they are collinear or am I missing a step.





2) Write down parametric vector equations for:


the line that goes through the point (1,3,2) and is orthogonal to





x= (3,-1,2) + t(6,0,2) AND


x= (3,4,5) + t(2,2,4)





For this question my answer is


x = (1,3,2) + t(1,5,-3)





Is this answer right? Also exactly how to get this answer, can somebody show me their working beacuse I only got the direction part (the second part) of the vector through trial and error.

Vectors???
1) It's enough to show that two of these vectors are parallel (scalar multiple of each other): CD, DE, CE. So I guess you did that.





2) Yes, the answer is right. Since you know that the line must go through (1,3,2), it is just right for the answer to be


x = (1,3,2) + t(1,5,-3)


Well, the answer could be anything like


x = (1+k, 3+5k, 2-3k) + t(1,5,-3) for any k and it would still be correct.


Getting the direction was perfectly enough, then you can just put the point that you know the line must go through in for the first part.





3) Yes, orthogonal means dot product is 0. That's correct how you do it!