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!

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- Feb 9th 2008, 11:55 AMoverduexIB Vectors
**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! - Feb 9th 2008, 12:13 PMearboth
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:

$\displaystyle \vec x = \left(\begin{array}{c}x\\y\end{array}\right) = \left(\begin{array}{c}2\\-1\end{array}\right) + r \cdot \left(\begin{array}{c}5\\-3\end{array}\right)~,~r \in \mathbb{R}$ - Feb 9th 2008, 12:23 PMmr fantastic
$\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.