Mian Systems LLC
Vector-Valued Functions or Vector Function (Contd.)
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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.
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.
What is the domain of a vector function?
These are all the possible values that variables are allowed to hold as place holders and the equation will hold true.

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)
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Solve for Unit Tangent Vector
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,
Solve: Next example is about finding parametric equation of tangent line
Solve-Find parametric equations for the tangent line