How to find the domain of a vector function
What is the domain of a vector function?
To find the domain of a vector function,
???r(t)=\langle a,b,c\rangle???
or
???r(t)=a\bold i+b\bold j+c\bold k???
we’ll need to find the domain of the individual components ???a???, ???b??? and ???c???.
Hi! I'm krista.
I create online courses to help you rock your math class. Read more.
Then the domain of the vector function is the values for which the domains of ???a???, ???b??? and ???c??? overlap.
How to find the domain of a vector function
Take the course
Want to learn more about Calculus 3? I have a step-by-step course for that. :)
Finding the domain of the vector function by looking at the domain of each component
Example
Find the domain of the vector function.
???r(t)=\left\langle\ln{(t-1)},t^2,\frac{1}{9-t^2}\right\rangle???
We need to find the domain of each component individually.
The domain of ???\ln{(t-1)}??? is
???t-1>0???
???t>1???
The domain of ???t^2??? is
all real numbers
The domain of ???\frac{1}{9-t^2}??? is
???9-t^2\neq0???
???-t^2\neq-9???
???t^2\neq9???
???t\neq\pm3???
Then the domain of the vector function is the values for which the domains of a, b and c overlap.
The first component restricts the domain of the entire vector function to ???t>1???. The second component doesn’t restrict the domain at all. The third component tells us that the domain can’t include ???t=\pm3???. ???t=-3??? is outside of the domain ???t>1???, so we can ignore it. But ???t=3??? is inside the domain ???t>1???, so we have to include it.
The domain of the vector function is
???t>1??? but ???t\neq3???
