calc 2 parametrization for a curve

Question

Find a parametrization for the curve described below.

the line segment with endpoints (-2,-4) and (-7,1)

Answer 1

Let A ( -2,-4) and B ( -7,1)

The directing vector u = AB =<- 5,5>

Therefore the parametrization of the line segment AB is as follows

r(t) = < -2 -5t, -4 +5t > , 0 ≤ t ≤ 1.

Now if we wish to obtain the arclength parametrization

we need r' (t ) = dr/ dt = < -5, 5 >

Then ds /dt = || r' (t ) || = √5²+5²

ds /dt = 5√2 This implies that with t as a parameter the speed on the curve is 5√2

ds= 5√2 dt

∫ds = ∫ 5√2 dt

s = 5√2 t then t = s /( 5√2 )

Then the arclength parametrization of the " curve "

is r (s ) = < < -2 -5s /( 5√2 ), -4 +5s /( 5√2 ) >

r (s ) = < < -2 -s /( √2 ), -4 +s /( √2 ) >

With the arclength parametrization the velocity

r'(s) = < - 1/( √2 ), - 1/( √2 > and the speed is always one

||dr(s)/ds|| = √(1/2 +1/2) =1