Vector problems I am unsure of

[Cuntryboner]

Hi all who read, need help with the following...
1. Two small boats A and B move so that at a time hours their position vectors are given by and where and are unit vectors of a magnitude 1km in the directions of east and north respectively.
Is it true that
A. The two boats collide after 3 hours.
B. The two boats collide after 2 hours.
C. The two boats don't collide.
D. The tow boats touch each other after two and 3 hours.
E. The two boats travel on straight line paths.

I under stand how to prove A and B are incorrect, but unsure of C, D and E. C is the correct response


2.  If P, Q and R are points with an origin O and  given the non-zero vectors , and . If where is a real constant then which of the following statements must be true.
A. P, Q and R are all linearly independant.
B. P, Q and R are all linearly dependant.
C. P, Q and R all lie on a straight line.
D. are linearly independant.
E. are linearly dependant.

Apparently C. is the best response.


3. If , then which of the following is FALSE .
A. P, Q and R are co linear
B.
C.
D.
E.
I am unsure how to disprove the solution, which is C.
Thanks for your help!!!

[kamil9876]

1.)

C.) For collision you need to find the values of t such that



and this happens if and only if both AND .

So you can show that they don't collide by showing that these sets of equations have no common solutions.

D.) isn't this just A and B? (or maybe there's a different between colliding and touching which has something to do with hitting each other at a tangent lol, but I'm not sure if that's what is meant)

E.) you can show this by showing that the cartesian equation is indeed a quadratic or square root curve. e.g for boat A and so plug it into .

2.)

The equation tells you that and are parralel. This must mean the three points are all on a striaght line.

Can you find examples where the other statements don't hold?

3.)



you know by the truth of B that |PQ| and |PR| are both 1.  as you can see by drawing. so the dot product is .

another way to prove it is:

[Cuntryboner]

Cheers, I see the logic in questions 1 and 3. I will get back to you on question 2:)