Question

This is Calculus 3; can anyone help me?
When two objects travel through space along two different curves, it is often important to know if they will collide. Suppose the trajectories of the two objects are given by the vector functions:
r1(t)=<-2+3t,-4+4t,2+t>
r2(t)= <2t,2t,8/t>
Will the two objects ever collide? If so, for what value of t will this happen?

Answer 1

each element of the position vector denotes the x,y,z values. Thus if the two objects collide their x,y,z values should be similar. Assuming the objects will collide, then,
$$r_1(t)=r_2(t)$$
$$<-2+3t,-4+4t,2+t>=<2t,2t,\frac{8}{t}>$$

$$-2+3t=2t \implies t=2$$
$$-4+4t=2t \implies t=2$$
$$2+t=\frac{8}{t}\implies t^2+2t-8=0\;(t\not=0) \implies (t+4)(t-2)=0 \implies t=2\; or\; t=-4$$

So each x,y,z positions are equal when $$t=2$$.
Thus we can conclude that two objects collide when t=2