# Calc III/Vector Calculus problems (moving particles in space curves)

• Feb 21st 2011, 01:55 PM
Chaobunny
Calc III/Vector Calculus problems (moving particles in space curves)
"Two particles travel along the space curves
$\displaystyle r_1(t)=<t,t^2,t^3>$ and $\displaystyle r_2(t)=<1 + 2t, 1+6t, 1+14t>$
Do they collide, and do their paths intersect?"

I calculated that $\displaystyle \dot{r_1}(t)=<1, 2t, 3t^2>$ and $\displaystyle \dot{r_2}(t)= < 2, 6, 14 >$ but I'm not entirely sure where to go from there or how I can use this information to solve the problem. I'm guessing I should find a t where all x coordinates and y coordinates and z coordinates are equal for the intersecting part? But how do I determine if they collide?

Edit: I'm also learning about shapes in 3D (hyperboloids, paraboloids, etc.) While I'm pretty good at determining what the shape will be based on the equation, I'm having trouble visualizing and drawing the graphs of difficult ones like hyperbolic paraboloids. Does anyone know of a (free) program that will allow me to graph functions of 2 variables so I can visualize this better? Thanks!
• Feb 22nd 2011, 03:34 AM
tonio
Quote:

Originally Posted by Chaobunny
"Two particles travel along the space curves
$\displaystyle r_1(t)=<t,t^2,t^3>$ and $\displaystyle r_2(t)=<1 + 2t, 1+6t, 1+14t>$
Do they collide, and do their paths intersect?"

These two questions are in fact one and the same, and what you've to do is to find out whether

there exist $\displaystyle t_0,s_0\in\mathbb{R}\,\,s.t.\,\,(t_0,t_0^2,t_0^3)= (1+2s_0,1+6s_0,1+14s_0)$ .

As far as I can see, vectorial derivatives have nothing necessarily to do with this question.

Tonio

I calculated that $\displaystyle \dot{r_1}(t)=<1, 2t, 3t^2>$ and $\displaystyle \dot{r_2}(t)= < 2, 6, 14 >$ but I'm not entirely sure where to go from there or how I can use this information to solve the problem. I'm guessing I should find a t where all x coordinates and y coordinates and z coordinates are equal for the intersecting part? But how do I determine if they collide?

Edit: I'm also learning about shapes in 3D (hyperboloids, paraboloids, etc.) While I'm pretty good at determining what the shape will be based on the equation, I'm having trouble visualizing and drawing the graphs of difficult ones like hyperbolic paraboloids. Does anyone know of a (free) program that will allow me to graph functions of 2 variables so I can visualize this better? Thanks!

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• Feb 22nd 2011, 04:19 AM
earboth
Quote:

Originally Posted by Chaobunny
"Two particles travel along the space curves
$\displaystyle r_1(t)=<t,t^2,t^3>$ and $\displaystyle r_2(t)=<1 + 2t, 1+6t, 1+14t>$
Do they collide, and do their paths intersect?"

I calculated that $\displaystyle \dot{r_1}(t)=<1, 2t, 3t^2>$ and $\displaystyle \dot{r_2}(t)= < 2, 6, 14 >$ but I'm not entirely sure where to go from there or how I can use this information to solve the problem. I'm guessing I should find a t where all x coordinates and y coordinates and z coordinates are equal for the intersecting part? But how do I determine if they collide?

Edit: I'm also learning about shapes in 3D (hyperboloids, paraboloids, etc.) While I'm pretty good at determining what the shape will be based on the equation, I'm having trouble visualizing and drawing the graphs of difficult ones like hyperbolic paraboloids. Does anyone know of a (free) program that will allow me to graph functions of 2 variables so I can visualize this better? Thanks!

1. If you solve for t

$\displaystyle r_1(t)=r_2(t)$

then you'll see that there isn't any value of t which satisfies this equation. Thus the particles doesn't collide.

2. If you solve for (s, t)

$\displaystyle r_1(t)=r_2(s)$

you'll get 2 valid solutions: $\displaystyle (s,t) = (0, 1)~\vee~(s,t)=\left(\frac12,\ 2\right)$

That means the paths of the two particles intersect in 2 different points. Plug in the values for (s, t) to get the coordinates of the points of intersection.