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Calculus IIChapter 6: 3-Dimensional SpaceSection 6-6

Vector Functions — Question 8

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Question 8

Curves C1C_1 and C2C_2 are given by r→1(t)=⟨t,t2,t3⟩,r→2(s)=⟨1−s,(1−s)2,s−1⟩.\vec r_1(t)=\left\langle t,t^2,t^3\right\rangle,\qquad \vec r_2(s)=\left\langle 1-s,(1-s)^2,s-1\right\rangle. Determine all geometric intersection points. If particles use their displayed parameters as time, determine whether they collide.

Original worksheet page 1: question and worked solution for 6-6-008
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Question 8 – Solution

Strategy Geometric intersection permits t≠st\ne s; collision requires t=st=s as well as equal positions.

See the diagram in the original worksheet below.

Intersection From the first coordinate, set u=1−s=tu=1-s=t. The second coordinates then agree automatically. The third coordinates require t3=−u=−tt^3=-u=-t, so t(t2+1)=0t(t^2+1)=0. Thus t=0t=0 and s=1s=1, giving the sole intersection (0,0,0)\boxed{(0,0,0)}.

Collision The required times differ, so particles using the displayed parameters do not collide.

Check At t=0t=0, r→1=0→\vec r_1=\vec 0; at s=1s=1, r→2=0→\vec r_2=\vec 0.

Original worksheet page 2: question and worked solution for 6-6-008

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