# Calculus 1: Direction and speed of vectors

It's long time I've had vectors. My friend asked me to help with this exercise given below:

The train stations for each of the following cities $$A$$, $$B$$, $$C$$, $$D$$ have the following coordinates: $$A=(-1,-2)$$, $$B=(10,3)$$, $$C=(1,5)$$ and $$D=(7,-1)$$ in a coordinate system. It is known that the $$X$$-axis is east and $$Y$$-axis is north. The distance between the cities are measured in kilometers. The points $$A$$ and $$B$$ are connected in a straight line and the points $$C$$ and $$D$$ are connected in a straight line. Both lines are intersecting at the point $$E$$. See the following image:

http://puu.sh/CWtnJ/cfd9427e2f.png

a) Compute unit vectors for $$A$$ to $$B$$ and $$C$$ to $$D$$.

Two trains at the same time leave the railway stations in $$A$$ and $$C$$. The train from $$A$$ to $$B$$ runs at $$100\text{ km/h}$$ and the train from $$C$$ to $$D$$ runs at $$65\text{ km/h}$$

b) Specify a vector describing the direction and speed of the movement of the train from $$A$$ to $$B$$

c) Specify a vector describing the direction and speed of the movement of the train from $$C$$ to $$D$$

d) Specify a parameter representation for the straight-line movement of the train from $$A$$ to $$B$$

e) Specify a parameter representation for the straight-line movement of the train from $$C$$ to $$D$$

f) Determine the coordinates of $$E$$. Will the two trains hit each other?

So my work is:

a) $$\vec{e_1}=\binom{0.91036}{0.41381}$$ and $$\vec{e_2}=\binom{0.70710}{-0.70710}$$

b) I don't really understand this question very well. Can anyone give me a hint here?

c) This is similar to b) so if I can solve b) after the hint, I can do c)

d) I believe it should be like this: $$\binom{x}{y}=\binom{-1}{-2}+t\binom{11}{5}$$

e) I believe it should be like this: $$\binom{x}{y}=\binom{1}{5}+t\binom{6}{-6}$$

f) And here I know how to calculate the intersection, but I don't know if the trains would hit each other. The intersection point $$E$$ is $$E=(5.19,0.81)$$

Note: I want hints. Thanks in advance

• $\vec{e1}$ is not correct. It should be the unit vector corresponding to $(10 - (-1), 3 - 2) = (11, 1)$ – ab123 Mar 7 at 14:47
• I forgot the negative sign. A=(-1,-2). I correct it now. – Anders Jørgensen Mar 7 at 14:49

(b), (c) You need the vectors in the direction of the vectors $$\vec{AB}$$ and $$\vec{CD}$$ with magnitude equal to the respective speeds.

(d), (e) Notice that the train should move $$\color{blue}{speed \times time}$$ distance along the vector direction.

The parametric representation can be done in terms of the position $$(x, y)$$ in terms of the time elapsed, say $$t$$.

For the first train, in $$t$$ (measured in hours), position is $$(x, y) = (-1, -2 ) + 100 \times t \times {\text{unit vector in direction of \vec{AB}}}$$.

$$\implies (x, y)=(-1, -2)+(11, 5)\frac{100t}{\sqrt{121 + 25}}$$. Similarly calculate for (e).

(f) You can do this in two different ways -

First, you can equate the two parametric representations to find if there exists a solution $$t'$$ (Equate the $$x$$ and $$y$$ coordinates and see if a solution of $$t$$ is possible). This corresponds to the trains reaching a point in space at the same time.

Second, if you have already found the point of intersection, calculate the value of time $$t$$ for both the trains. If they are the same, trains meet at a point, otherwise, they don't.

• Good, thank you for the hints. Regarding (b) and (c), if I find the magnitude and direction, then I have found direction and speed? – Anders Jørgensen Mar 7 at 15:14
• Yes, I think the magnitude of that vector must be equal to the speed. Edited that in the answer – ab123 Mar 7 at 15:16
• For (f) I did the calculations in Maple and obtained different values of $t$. I also try with differents speed (extra exercise) with 110km/h and 95km/h and they didn't hit either. – Anders Jørgensen Mar 7 at 15:38
• Ohh, I did a mistake. They actually hit in the point E. Also for 110km/h and 95km/h – Anders Jørgensen Mar 7 at 16:04