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
(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.