# Chapter 14 - Calculus of Vector-Valued Functions - 14.1 Vector-Valued Functions - Exercises - Page 709: 4

$P$ does not lie on the path $Q$ lies on the path

#### Work Step by Step

For the point $P=(4,11,20)$, we have the system $$1+t=4, \quad 2+t^2=11, \quad t^4=20$$ which has no solutions since the first equation gives $t=3$ which is not a solution for the third equation. Hence, $P$ does not lie on the given path. For the point $Q=(-1,6,16)$, we have the system $$1+t=-1, \quad 2+t^2=6, \quad t^4=16$$ which has the solution $t=2$. Hence, $Q$ lies on the given path.

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