   Chapter 13.1, Problem 50E

Chapter
Section
Textbook Problem

# Two particles travel along the space curvesr1 (t) = ⟨t, t2, t3⟩ r2 (t) = ⟨1 + 2t, 1 + 6t, 1 + 14t⟩Do the particles collide? Do their paths intersect?

To determine

To find: Whether the vector functions of particles collide and whether paths of particles intersect.

Explanation

Given data:

The vector functions of two particles as r1(t)=t,t2,t3 and r2(t)=1+2t,1+6t,1+14t .

Equate the components of vector equations of two particles and find the value of parameter t. If the determined value of t satisfies all the equated expressions, then the particles collide at that point.

Equate the vector functions of two particles.

r1(t)=r2(t)

Substitute t,t2,t3 for r1(t) and 1+2t,1+6t,1+14t for r2(t) ,

t,t2,t3=1+2t,1+6t,1+14t

Equate each component.

t=1+2t (1)

t2=1+6t

t26t1=0 (2)

t3=1+14tt314t1=0

Solve equation (1).

2tt+1=0t+1=0t=1

Therefore, the value of t is either –1.

Substitute –1 for t in equation (2),

(1)26(1)1=01+61=060

The value of LHS is not equal to RHS. Hence equation is not satisfied at t=1 .

Therefore, the particles do not collide at t=1 .

Equate the components of equation r1(t)=r2(s) and find the value of s. If determined value of s satisfies the equations, then the particle do intersect at that point.

Equate the vector functions r1(t)=r2(s) .

t,t2,t3=1+2s,1+6s,1+14s

Equate the components on both sides of equation

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