How to find the value of tangent vectors?

Question

In the figure $z_1=a_1+ib_1$ and $z_2=a_2+ib_2$ are the two curves. $T_1$ and $T_2$ be the tangents on the curves $z_1$ and $z_2$. What I am interested to know what will be the tangent vectors?

Answer 1

The tangent vectors will be: $$\frac{dz_1}{dt}=\frac{da_1}{dt}+\frac{dib_1}{dt}=\frac{da_1}{dt}+0 \cdot b_1 +i\frac{db_1}{dt}=\frac{da_1}{dt} +i\frac{db_1}{dt}$$ $$\frac{dz_2}{dt}=\frac{da_2}{dt}+\frac{dib_2}{dt}=\frac{da_2}{dt}+0\cdot b_2 +i\frac{db_2}{dt}=\frac{da_2}{dt} +i\frac{db_2}{dt}$$ Is this what you were looking for?

Since $(a'_1,b'_1) \cdot (a'_2,b'_2)=|(a'_1,b'_1)||(a'_2,b'_2)|\cos{\alpha}$: $$\alpha=\cos^{-1}(\frac{(a'_1,b'_1) \cdot (a'_2,b'_2)}{|(a'_1,b'_1)||(a'_2,b'_2)|})$$ Given $(a'_1,b'_1),(a'_2,b'_2) \neq (0,0)$