About is tangent vector?

Question

Vector equation for a line is like this $$r(t)=r_0+tv$$

But I wonder, when $tv$ is like $$\langle t,\frac{-3t}{4},\frac{3t}{2}\rangle$$ then can I make it look better like $$\langle4t, -3t, 6t\rangle$$

Answer 1

Yes of course any non zero multiple of $v=\left(1,\frac{-3}{4},\frac{3}{2}\right)$ can be used as direction vector for the given line.

Indeed just assume a scaling factor for the parameter as for example $t=4\bar t$ to obtain

$$tv=4\bar t v=\bar t (4v)$$

and the two lines with $t,\bar t \in \mathbb{R}$

• $r(t)=r_0+tv$
• $\bar r(t)=r_0+\bar t(4v)$

decribe, with different parametrization, the same set of points in $\mathbb{R^3}$.