# Domains Find the domain of the following vector-valued function. r\left(t\right)=\frac{2}{t-1}i+\frac{3}{t+2}j

Domains Find the domain of the following vector-valued function.
$$\displaystyle{r}{\left({t}\right)}={\frac{{{2}}}{{{t}-{1}}}}{i}+{\frac{{{3}}}{{{t}+{2}}}}{j}$$

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Data analysis
To find the domain of the vector valued function.
Given vector function,
$$\displaystyle{r}{\left({t}\right)}={\frac{{{2}}}{{{t}-{1}}}}{i}+{\frac{{{3}}}{{{t}+{2}}}}{j}$$
Domain of vector function
The domain of a vector function is the values of vector function parameter for which the components of vector function are defined.
Here, 't' is the function parameter and coefficients of i and j are the components.
Horizontal component is defined if,
$$\displaystyle{\frac{{{2}}}{{{\left({t}-{1}\right)}}}}$$ is defined for all the values of 't' in real excluding $$\displaystyle{t}={1}$$.
As when $$\displaystyle{t}={1}$$, component ($$\displaystyle{\frac{{{1}}}{{{2}}}}$$) which is undefined.
Vertical component is defined if,
$$\displaystyle{\frac{{{3}}}{{{\left({t}+{2}\right)}}}}$$ is defined for all the values of 't' in real excluding $$\displaystyle{t}=-{2}$$.
As when $$\displaystyle{t}=-{2}$$, component $$\displaystyle={\frac{{{3}}}{{{0}}}}$$ which is undefined.
Hence, Domain of $$r\left(t\right)=\left\{t\mid R-\left\{-2, 1\right\}\right\}$$
Where R is real numbers.