Find the Cartesian equation corresponding to the following vector equation: $r(t) = \frac{(e^t + e^{-t})}{2} \hat i + +\frac{(e^t - e^{-t})}{2}\hat j$

Question

Unsure how to go about converting this from vector equation to cartesian. Any help much appreciated, thanks!

Answer 1

Your $x$ component is $\cosh t$ and $y$ component is $\sinh t$ and as $\cosh^2 t-\sinh^2 t=1$ ,so its equation is $x^2-y^2=1$.

Answer 2

Hint: We are looking for an equation stating the connection between the coordinates of $r(t)$.
Note that $a=e^t$ can be any positive number.

So, take $x=\frac{a+1/a}2$ and $y=\frac{a-1/a}2$, and calculate $x^2-y^2$.
Also, remember $x>0$.