1. ## Simple parametrization problem

How do I parametrize a straight line segment from (2,-1,3) to (4,2,-1)?

How do I parametrize path C from (0,0,0) to (4,2,3) that consists of three line segments parallel to x, y and z-axis in that order?

How do I parametrize path C from (-1,2,-2) to (1,5,2) that consists of three line segments parallel to the z-axis, x-axis, and y-axis in that order?

These are probably simple but I really need help

2. When we move from $\displaystyle (2,-1,3)$ to $\displaystyle (4,2,-1)$, the following changes take place:

$\displaystyle x$ increases $\displaystyle 2$ units
$\displaystyle y$ increases $\displaystyle 3$ units
$\displaystyle z$ decreases $\displaystyle 4$ units

Now let's say we want the independent variable of the parametrized line, $\displaystyle t$, to range from $\displaystyle 0$ to $\displaystyle 1$, so that as $\displaystyle t$ moves from $\displaystyle 0$ to $\displaystyle 1$, the point $\displaystyle (x(t),y(t),z(t))$ will move from $\displaystyle (2,-1,3)$ to $\displaystyle (4,2,-1)$. In that case, since

$\displaystyle (x,y,z)=(2,-1,3)$

at $\displaystyle t=0$, we obtain for our parametrization

$\displaystyle (x,y,z)=(2+2t,-1+3t,3-4t).$

Hint for the next problem: the first line of the path is given by

$\displaystyle (x,y,z)=(4t,0,0).$