Vector equation of a straight line passing through a given point and parallel to a given vector
Let $O$ be the origin, $A$ be the given point and $\overrightarrow{MN}=\overrightarrow{b}$ be the given vector. Let $P$ be any point on the line through $A$ and parallel to the given vector $\overrightarrow{b}$. Let $\overrightarrow{OP}=\overrightarrow{r}$.
Since $\overrightarrow{AP}$ and $\overrightarrow{MN}$ are parallel vectors, so \[\overrightarrow{AP}= t\:\overrightarrow{MN}=t\:\overrightarrow{b}\] where $t$ is a scalar. Using triangle law of vector addition, \[\overrightarrow{OP}=\overrightarrow{OA}+\overrightarrow{AP}\] \[\overrightarrow{r} =\overrightarrow{a}+t\:\overrightarrow{b}\]
Since the different values of $t$ will give the position vectors of different points on the line $AP$, so the relation is true for any point on the line $AP$. Hence, it represents the vector equation of a straight line.
Cor. If the line passes through the origin, then $\overrightarrow{a}=(0,0)$, so \[\overrightarrow{r}=t\: \overrightarrow{b}\]
Vector equation of a straight line passing through two given points
Let $O$ be the origin and let $A$ and $B$ be two given points. Let $\overrightarrow{OA}= \overrightarrow{a}$ and $\overrightarrow{OB}=\overrightarrow{b}$.
Then, by using triangle law of vector addition in $\Delta OAB$, \[\overrightarrow{OA}+\overrightarrow{AB} =\overrightarrow{OB}\] \[\overrightarrow{AB}=\overrightarrow{OB}-\overrightarrow{OA}= \overrightarrow{b}-\overrightarrow{a}\]
Let $P$ be any point on the line through $A$ and $B$. Let $\overrightarrow{OP}=\overrightarrow{r}$.
Since $\overrightarrow{AP}$ and $\overrightarrow{AB}$ are parallel vectors, so \[\overrightarrow{AP}= t\:\overrightarrow{AB}=t(\:\overrightarrow{b}-\overrightarrow{a}\:)\] where $t$ is a scalar.
Using triangle law of vector addition in $\Delta OAP$, \[\overrightarrow{OP}=\overrightarrow{OA}+ \overrightarrow{AP}\] \[=\overrightarrow{a}+t(\:\overrightarrow{b}-\overrightarrow{a}\:)\] \[=(1-t)\: \overrightarrow{a}+t\:\overrightarrow{b}\] which represents the vector equation of a straight line through $A$ and $B$.
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