Velocity as a Tangent Vector

Part 2: Velocity as a Tangent Vector

If r( t) is the position vector of a moving object at time t, then the vector r( t+Dt) -r( t) is a secant vector whose initial and terminal points are on the curve traced by the terminal end of r( t) .

Thus, as Dt approaches 0, the resulting secant vectors more and more resemble the curve itself in the vicinity of r( t) ,

so that as Dt approaches 0, the difference quotient

r( t+Dt) -r( t)

approaches the tangent vector shown in red above. Since the difference quotient also approaches v( t) as Dt approaches 0, we obtain the following theorem:

Theorem 6.2: The velocity vector v = dr/dt is tangent to the parameterized curve at the point with position r( t) .

A curve which has a tangent vector at each point is said to be smooth, and a parameterization r(t), t in [a,b], is said to be smooth if it parameterizes a smooth curve. Specifically, r(t), t in [a,b], is smooth if v(t) exists and is nonzero for all t in [a,b].

EXAMPLE 3    Sketch v(1) at the point of tangency to r( t) = át³,t² ñ .
Solution: To do so, we first differentiate to obtain

v( t) =

dr

dt

= á3t²,2t ñ

We then let t = 1 to obtain

v( 1) = á 3,2 ñ

which when graphed versus r( t) yields the following:

EXAMPLE 4    Sketch v( 5p/2) at the point of tangency to the helix r( t) = á 4cos( t), 4sin( t) ,t ñ .
Solution: To do so, we notice that

v( t) = á -4 sin( t), 4 cos(t) ,1 ñ

so that the velocity vector at t = 5p/2 is

v æ
è

5p

2

ö
ø = -4 sin æ
è

5p

2

ö
ø , 4 cos æ
è

5p

2

ö
ø ,1 = á -4,0,1 ñ

which is graphed along with the curve below:

Check your Reading: Which plane is v( p) parallel to if r( t) = á cos( t) ,sin( t) ,t ñ ?