# Math Insight

### Parametrization of a line examples

#### Example 1

Find a parametrization of the line through the points $(3,1,2)$ and $(1,0,5)$.

Solution: The line is parallel to the vector $\vc{v}= (3,1,2) - (1,0,5) = (2,1,-3)$. Hence, a parametrization for the line is \begin{align*} \vc{x} = (1,0,5) + t(2,1,-3) \qquad \text{for} \quad -\infty < t < \infty. \end{align*} We could also write this as \begin{align*} \vc{x} = (1+2t, t, 5-3t) \qquad \text{for} \quad -\infty < t < \infty. \end{align*} Or, if we write $\vc{x}=(x,y,z)$, we could write the parametric equation in component form as \begin{align*} x &= 1+2t,\\ y &= t,\\ z &= 5-3t, \end{align*} for $-\infty < t < \infty$.

#### Example 2

Find a parametrization for the line segment between the points $(3,1,2)$ and $(1,0,5)$.

Solution: The only difference from example 1 is that we need to restrict the range of $t$ so that the line segment starts and ends at the given points. We can parametrize the line segment by \begin{align*} \vc{x} = (1,0,5) + t(2,1,-3) \qquad \text{for} \quad 0 \le t \le 1. \end{align*} (Or we could use any of the other forms of the parametric equation in example 1, as long as we restrict $t$ to lie in the interval $[0,1]$.)