IB Vectors

• February 9th 2008, 11:55 AM
overduex
IB 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 :)
• February 9th 2008, 12:13 PM
earboth
Quote:

Originally Posted by overduex
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.

Vector $\vec a$ points at the point A and $\vec b$ points at the point B:

$\vec a=\left(\begin{array}{c}2\\-1\end{array}\right)$

$\vec b=\left(\begin{array}{c}-3\\2\end{array}\right)$

The direction of the line AB is determined by $\vec a - \vec b = \left(\begin{array}{c}5\\-3\end{array}\right)$

The line AB passes either through A or B. Therefore the equation of the line is:

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

Originally Posted by overduex
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 :)

$\vec{OA} = \left( \begin{array}{c}
2 \\
-1 \end{array} \right)
$
and $\vec{OB} = \left( \begin{array}{c}
-3 \\
2 \end{array} \right)
$
.

$\vec{AB} = \vec{AO} + \vec{OB} = -\vec{OA} + \vec{OB} = - \left( \begin{array}{c}
2 \\
-1 \end{array} \right) + \left( \begin{array}{c}
-3 \\
2 \end{array} \right) = \left( \begin{array}{c}
-2 \\
1 \end{array} \right) + \left( \begin{array}{c}
-3 \\
2 \end{array} \right)$

$= \left( \begin{array}{c}
-5 \\
3 \end{array} \right)$
.

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.