Please help me with this problem given for test practice! THANK YOU SO MUCH!! 3.Two particles travel along the space curves r; (t) = (t, t²,t³) and r2 (t) = (1+ 2t, 1 + 6t, 1 + 14t), do their paths intersect? Do the particles collide? If so, determine the point(s) of intersection.

Given data: The first curve is r1(t)=<t, t2, t3>. The second curve is r2(t)=<1+2t, 1+6t,…

Answered: Two particles travel along the space…

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter11: Topics From Analytic Geometry

Section11.4: Plane Curves And Parametric Equations

Problem 34E

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Please help me with this problem given for test practice! THANK YOU SO MUCH!!

3.
Two particles travel along the space curves r; (t) = (t, t²,t³) and r2 (t) = (1+ 2t, 1 + 6t, 1 + 14t), do their

paths intersect? Do the particles collide? If so, determine the point(s) of intersection.

Expert Solution

Step 1

Given data:

The first curve is r₁(t)=<t, t², t³>.

The second curve is r₂(t)=<1+2t, 1+6t, 1+14t>.

The expression to determine whether they collide or not is,

$\begin{array}{rcl} r_{1} (t) & = & r_{2} (t) \\ <t, t^{2}, t^{3}> & = & <1 + 2 t, 1 + 6 t, 1 + 14 t> \\ t & = & 1 + 2 t . . . . . . . . . . . . . . . . . . . . (I) \\ t^{2} & = & 1 + 6 t . . . . . . . . . . . . . . . . . . . (II) \\ t^{3} & = & 1 + 14 t . . . . . . . . . . . . . . . . . . (III) \end{array}$

Evaluate the value of t from equation(I).

$\begin{array}{rcl} t & = & 1 + 2 t \\ t & = & - 1 \end{array}$

Substitute -1 for t in equation(II) and equation(III).

$\begin{array}{rcl} {(- 1)}^{2} & = & 1 + 6 (- 1) . . . . . . . . . . (II) \\ 1 & \neq & - 5 \\ {(- 1)}^{3} & = & 1 + 14 (- 1) . . . . . . . . . (III) \\ - 1 & \neq & - 13 \end{array}$

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