### 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 $[0,1]$.)

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