Calculus 3 particles

Question

A particle moves at a constant speed along a line through P=(4,5,1) to point Q=(9,3,−1). Find the parametric equations and interval for its potions at time t if the particle passes through P at time t=0 and has a constant speed of 3.Give exact answers or round to at least 5 decimal places.

Answer 1

The directing vector of the line passing through the points P and Q is u=<5,-2,-2>

then the parametric equations of the line with respect to t ( time ) are

x = 4 + 5t

y = 5 −2t

z = 1 −2t

0 ≤ t ≤ 1

But since the speed is 3=s / t ⇒ t= s /3 where s denotes the distance traveled.

Then a repametrization is in order

x = 4 + 5( s /3 )

y = 5 −2( s /3)

z = 1 −2( s /3)

0 ≤s ≤3

Then finally

x = 4 + (5/3)s

y = 5 − (2/3)s

z = 1 − (2/3)s

0 ≤s ≤3