Chapter 14

Section 14.3Curves

The Direction of the Tangent VectorTo see the geometric meaning of the derivative of a vector function consider the curve and where you are located at a time . Remember the derivative is the following limit.

r ( 𝑡0 )r ( 𝑡0+h )−r ( 𝑡0 )

r (𝑡 0+ h2 )− r ( 𝑡0 )

r ′ (𝑡 0 )

x

y

z

The direction the derivative points in at time is a vector tangent to the curve that has the same orientation (direction of travel) as the curve itself. This can be used to find the vector function for the tangent line to the curve.

ExampleFind the equation of the tangent line to the curve given to the right at time .

r (𝑡 )= ⟨7 𝑡−5cos (𝑡−3 ) ,− 𝑡2 , ln (𝑡−2 ) ⟩

To get the equation of the line we need to know two things, a direction vector d and a point on the line P. To find d find the derivative and plug in 3 in the derivative. To find P plug 3 into the original function.

d=r ′ (3 )= ⟨7 ,−6,1 ⟩r ′ (𝑡 )=⟨7+5sin (𝑡−3 ) ,−2 𝑡 , 1𝑡−2 ⟩ 𝑃 : r (3 )=⟨16 ,−9,0 ⟩

The equation of the tangent line is:

The Length of the Tangent VectorLet the variable be the distance traveled along the curve from time to time , this is called the arc length. As gets smaller the length of the vector and the value s (arc length) get closer.

r ( 𝑡0 )r ( 𝑡0+h )−r ( 𝑡0 )

x

y

z 𝑠=arc   length

ExampleAll three curves below describe a circle with center at the origin traveling counterclockwise around it. Find the time to complete 1 lap, average rate of speed, and .

𝑥2+ 𝑦2=4

1 lap Average rate

Distance for 1 lap is

1 lap Average rate

1 lap Average rate

Motion, Velocity vector and SpeedThe derivative of the position vector, is called the velocity vector. The length of this vector is called the speed. Remember velocity is a vector giving direction speed is a scalar.

Unit Tangent VectorTo measure only the change in the direction of a curve we use the unit Tangent vector T. This is a unit vector pointing in the direction of the velocity vector.

Since then , now take the derivative of both sides of this equation. From this we see that the unit tangent is always perpendicular to its derivative.

Unit Normal VectorIf we take the derivative of the unit tangent vector and make it of length 1 point in the same direction we call this the unit Normal vector N. This is a unit vector perpendicular to the unit tangent vector.

T (𝑡 )

N (𝑡 )

ExampleFind the velocity, speed, unit tangent and unit normal vectors for the curve given to the right. r (𝑡 )= ⟨𝑡 3+1,5,3 𝑡−4 ⟩Velocity = Speed = Unit Tangent = Unit Normal =

ExampleThe two curves to the right intersect at the point . Find the cosine of the angle of intersection between the curves (i.e. angle between tangent lines).

The curves intersect at and compute tangent vectors and dived by their lengths.

and

and

cos𝜃=⟨2,4,0 ⟩ ∙ ⟨1,0,1 ⟩

‖⟨2,4,0 ⟩‖‖⟨1,0,1 ⟩‖= 2

√20√2= 1

√10