I am tackling with this boat problem.
Two boats, A and B, move so fast that at time t hours, their position vectors, in kilometers, are rA=(9t)i+(3-6t)j, rB=(7-4t)i+(7t-6)j
And problems are shown below:
(A)find the coordinates of the commmon point of the paths of the two boats. (B)Show that the boats do not collide.
I thought that finding the common point means equal the two vectors. So I lists two equations: 9t=7-4t, 3-6t=7t-6, but there comes two different results of t.
Maybe this can proves that boats do not collide, but how can I find their common point??
Help!!
The path of the first boat is the graph of $(9t,3-6t)$ as $t$ ranges through the real numbers. If $x = 9t$ and $y = 3-6t$, then $y = 3 - \frac{2}{3}x$. The path of the second boat can be found similarly. Now it's possible, regardless of the value of $t$, to find the intercept (x,y) of both paths. Showing that, if rA(u) = (x,y) and rB(v)=(x,y) then $u \neq v$ is enough to prove the boats dont collide.