Find the position vectors that join the origin to the points with coordinates A(2, -1) and B(-3, 2). Express your answers as column vectors. Hence find line AB.
All help is welcome
Thanks in advance!
Vector $\displaystyle \vec a$ points at the point A and $\displaystyle \vec b$ points at the point B:
$\displaystyle \vec a=\left(\begin{array}{c}2\\-1\end{array}\right)$
$\displaystyle \vec b=\left(\begin{array}{c}-3\\2\end{array}\right)$
The direction of the line AB is determined by $\displaystyle \vec a - \vec b = \left(\begin{array}{c}5\\-3\end{array}\right)$
The line AB passes either through A or B. Therefore the equation of the line is:
$\displaystyle \vec x = \left(\begin{array}{c}x\\y\end{array}\right) = \left(\begin{array}{c}2\\-1\end{array}\right) + r \cdot \left(\begin{array}{c}5\\-3\end{array}\right)~,~r \in \mathbb{R}$
$\displaystyle \vec{OA} = \left( \begin{array}{c}
2 \\
-1 \end{array} \right)
$ and $\displaystyle \vec{OB} = \left( \begin{array}{c}
-3 \\
2 \end{array} \right)
$.
$\displaystyle \vec{AB} = \vec{AO} + \vec{OB} = -\vec{OA} + \vec{OB} = - \left( \begin{array}{c}
2 \\
-1 \end{array} \right) + \left( \begin{array}{c}
-3 \\
2 \end{array} \right) = \left( \begin{array}{c}
-2 \\
1 \end{array} \right) + \left( \begin{array}{c}
-3 \\
2 \end{array} \right)$
$\displaystyle = \left( \begin{array}{c}
-5 \\
3 \end{array} \right)$.
Earboth is obviously a quicker hand at latex than me. Note that his/her vector and mine are both in the direction of the line .... just in opposite directions. It doesn't matter which one you use.