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\centerline
{
\bf Dr. Z's Math251 Handout \#13.1 [Vector Functions and Space Curves]
}

By Doron Zeilberger

{\bf Problem Type 13.1a}: Find a vector equation
and a parametric equation for the line segment
joining $P(p_1,p_2,p_3)$ and $Q(q_1,q_2,q_3)$

{\bf Example Problem 13.1a}: Find a vector equation
and a parametric equation for the line segment
joining $P(1,0,1)$ and $Q(2,3,1)$

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{\bf Steps} & {\bf Example} \cr  &  \cr
{\bf 1.} 
The vector equation for the line segment
joining $P$ and $Q$ is $(1-t)P+tQ$ ($0 \leq t \leq 1$),
i.e.
$$
(1-t) \langle p_1,p_2,p_3 \rangle+
t \langle q_1,q_2,q_3 \rangle
\quad , \quad (0 \leq t \leq 1) \quad .
$$
&
{\bf 1.} 
$$
(1-t) \langle 1,0,1 \rangle+t \langle 2,3,1 \rangle=
$$
$$
 \langle 1-t \, , \, 0 \, , \, 1-t \, \rangle+ 
\langle 2t \, , \, 3t \, , \, t \, \rangle=
 \langle \, 1+t \, , \, 3t \, , \, 1 \, \rangle  \quad ,
$$
$$
\quad (0 \leq t \leq 1) \quad.
$$
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{\bf 2.} After you simplify $tP+(1-t)Q$ read-off
the three components, $x$, $y$, $z$.
&
{\bf 2.} 
$$
 \langle \, x \, , \, y \, , \, z \, \rangle =
 \langle \, 1+t \, , \, 3t \, , \, 1 \, \rangle  \quad,
$$
So
$$
x=1+t \quad , \quad y=3t \quad ,\quad  z=1 \quad (0 \leq t \leq 1) \quad.
$$
This is the {\bf Ans.}.
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