# Simple parametrization problem

• Dec 5th 2009, 11:13 AM
messianic
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
• Dec 6th 2009, 06:33 AM
Scott H
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).\$