Vector-Valued Functions or Vector Function** (Contd.)**

As **t** increases, the terminal points of the red vector produced by the vector valued function trace out the blue curve.

As **t** increases, the terminal points on the red vector produced by the vector-valued function trace out the blue

curve.

curve.

In the next example we have a space curve because it has x,y, & z component.

Vector produced by vector-valued function when t = -8. As** t **increases, it will trace out the space curve and in an orientation where vector

is climbing up.

is climbing up.

r(t) = t-2/t+2ti +sintj + ln(9-t2)k

It can be written in component form

= < t-2/t+2t, sint, ln(9-t2)>

Looking for values that satisfies the equation

t-2/t+2t ;if t= -2, the the denominator will be zero. Hence, t cannot be -2

sint --> no restrictions on t

Ln(9-t2) 9-t > 0 ;can’t have negative in parenthesis for a natural log (ln) ln > 0

(3-t)(3+t)

t=+- 3 ; will not work

Use values between -3 and +3

Domain is (-3,-2) U (-2,3)

Step 1 Find tangent vector (take derivative)

Step 2 Plug value of ‘t’

Step 3 Unit tangent vector is finding magnitude and find unit vector u,

Step 2 Plug value of ‘t’

Step 3 Unit tangent vector is finding magnitude and find unit vector u,

Solve-Find parametric equations for the tangent line