Vector equation of a line : Maths Notes from Mr Shelswell

You should already be familiar with the equation of a straight line in a 2D plane as being $y = mx + c$ . This section takes that work a step further. It allows us to express the equation of a straight line in 3D space in a similar format.

Take a straight line through the point A with direction vector $\mathbf{p}$ . R is another point on that line. The position vector (from an arbitrary origin) of A is $\mathbf{a}$ , and of R is $\mathbf{r}$ . The vector $\overrightarrow{ \mathrm{AR} }$ is a multiple, $t$ , of $\mathbf{p}$ .

It follows that the position of a point R on the line is given by:
$\mathbf{r} = \overrightarrow{ \mathrm{OR} } = \overrightarrow{ \mathrm{OA} } + \overrightarrow{ \mathrm{AR} } = \mathbf{a} + t \mathbf{p}$

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2 July 2013, Created with GeoGebra